Terahertz Laser Radiation Induced Opto-Electronic Eﬀects ... · 2.1 Band Structure of Mercury Telluride HgTe belongs to the group of II-VI semiconductors and crystallizes in zinc-blende

Terahertz Laser Radiation Induced

Opto-Electronic Effects in HgTe Based

Nanostructures

DISSERTATION

zur Erlangung des Doktorgrades der Naturwissenschaften

doctor rerum naturalium

(Dr. rer. nat.)

Fakultat fur Physik

Universitat Regensburg

vorgelegt von

Dipl.-Phys. Christina Zoth

aus Gottingen

im Jahr 2014

Die Arbeit wurde von Prof. Dr. Sergey D. Ganichev angeleitet.

Das Promotionsgesuch wurde am 11. Juni 2014 eingereicht.

Prufungsausschuss:

Vorsitzender: Prof. Dr. Gunnar S. Bali

1. Gutachter: Prof. Dr. Sergey D. Ganichev

2. Gutachter: Prof. Dr. Christian Schuller

weiterer Prufer: Prof. Dr. Thomas Niehaus

Contents

1 Introduction 5

2 Theoretical Basics 8

2.1 Band Structure of Mercury Telluride . . . . . . . . . . . . . . . 8

2.2 Photogalvanic Effects . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Photoconductivity Effect . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Cyclotron Resonance . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 Quantum Transport Phenomena . . . . . . . . . . . . . . . . . . 18

3 Experimental Methods 22

3.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Investigated Samples . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.1 Mercury Telluride Quantum Well Samples . . . . . . . . 26

3.2.2 Diluted Magnetic Semiconductor Samples . . . . . . . . 30

3.3 Electrical and Optical Measurement Techniques . . . . . . . . . 32

3.3.1 Characterization via Magneto-Transport Measurements . 32

3.3.2 THz Radiation Induced Photocurrents . . . . . . . . . . 34

3.3.3 Measurement of Photoresistance . . . . . . . . . . . . . . 34

3.3.4 Measurement of Radiation Transmittance . . . . . . . . 35

3.4 Development and Characterization of THz Waveplates . . . . . 36

4 Magnetogyrotropic Photogalvanic Effect in a Diluted Mag-

netic Semiconductor Heterostructure 38

4.1 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3 Brief Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

CONTENTS 4

5 Dirac Fermion System at Cyclotron Resonance 46

5.1 Terahertz Induced Photogalvanic Effects . . . . . . . . . . . . . 46

5.1.1 Photocurrents at Zero Magnetic Field . . . . . . . . . . . 47

5.1.2 Cyclotron Resonance Assisted Photocurrents . . . . . . . 48

5.2 Photoresistance Measurements . . . . . . . . . . . . . . . . . . . 54

5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.3.1 Photocurrents at Zero Magnetic Field . . . . . . . . . . . 56

5.3.2 Photocurrents and Photoresistance under Cyclotron Res-

onance Condition . . . . . . . . . . . . . . . . . . . . . . 57

5.3.3 Microscopic Picture of Current Generation . . . . . . . . 62

5.4 Brief Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6 Quantum Oscillations in Mercury Telluride Quantum Wells 68

6.1 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . 68

6.1.1 Results for a Parabolic Dispersion . . . . . . . . . . . . . 69

6.1.2 Results for a Dirac Fermion System . . . . . . . . . . . . 75

6.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.2.1 Microscopic Picture of Current Generation . . . . . . . . 78

6.2.2 Analysis of the Photocurrents . . . . . . . . . . . . . . . 81

6.2.3 Analysis of the Photoresistance . . . . . . . . . . . . . . 86

6.3 Brief Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

7 Conclusion 90

References 92

1 Introduction

The spin of electrons and holes in solid state systems attracts growing atten-

tion in modern research. This is stimulated by the hope that the spin degree

of freedom in semiconductor systems can be utilized for new spin based elec-

tronics, namely spintronics [1, 2]. The success of spintronics depends on the

ability to create, manipulate and detect, as well as to transport and store spin

orientations [3]. Possibilities of spin manipulation are e.g., the application

of light or an electric field to a semiconductor heterostructure. A promising

physical effect which can be exploited for this kind of spin manipulation is the

spin orbit interaction [4, 5]. Therefore, semiconductors which are character-

ized by a strong spin orbit interaction form a suitable system for spintronic

applications [6].

A very promising candidate is mercury telluride, which has recently advanced

into the focus of condensed matter research. In particular, the strong spin orbit

interaction in mercury telluride is responsible for its unique energy spectrum

yielding very fascinating effects [7–13]. Herein, without changing the material

it is possible to obtain an inverted and normal parabolic band structure, as

well as a linear one. The latter state is described by the Dirac theory of

massless fermions [12]. In mercury telluride this Dirac-like dispersion can be

realized in different manners: either by layered quantum well structures with

a characteristic quantum well width or in topological insulator systems. The

main difference lies in the spin degeneracy of the Dirac cone and the topology of

the band structure in real space. In the former case the linear band structure is

formed by unpolarized two dimensional Dirac fermions. This state is not bound

to any spatial limitation [13–15]. In case of topological insulators the Dirac

states only emerge at the edges or on the surface of a two or three dimensional

mercury telluride system, respectively. Furthermore the states demonstrate

a spin polarization [10]. This gives rise to the existence of dissipationless

spin polarized transport along the one dimensional edge channels or the two

dimensional surface states, as back scattering with spin flip is prohibited [10,

12,16].

1 INTRODUCTION 6

An interesting and powerful access to the study of spin related phenomena is

provided by opto-electronic techniques. In particular, using radiation in the

terahertz range the observation of effects of carrier redistribution in momen-

tum space becomes possible e.g., photogalvanic effects [17]. Moreover, the

measurements of opto-electronic effects build a bridge between the two fields

of transport and optical phenomena [1]. The fundamental theory of these phe-

nomena e.g., photogalvanic effects, photoconductivity or quantum transport

are introduced in Chapter 2. Chapter 3 contains details about the experimen-

tal setup and gives an overview of the investigated samples.

The observation and study of the terahertz radiation induced magnetogy-

rotropic photogalvanic effect in various mercury telluride quantum well systems

forms the core of this thesis. Herein several new phenomena are observed. In

order to understand this opto-electronic and spin dependent effect, results for

a well established semiconductor system will be presented in advance as this

will substantially help to understand the observations for the mercury telluride

structures. For this purpose quantum well structures based on cadmium tel-

luride and indium arsenide with a diluted amount of magnetic manganese are

investigated [18,19]. The results will be illustrated and discussed in Chapter 4.

In Chapters 5 and 6 the results of the opto-electronic measurements on mercury

telluride based quantum well nanostructures are presented. In the scope of this

thesis, the first investigations of cyclotron resonance assisted photocurrents and

photoresistance on a Dirac fermion system are performed: the results which are

presented and discussed in Chapter 5 for a mercury telluride sample with the

critical quantum well width give experimental proof that the structure under

investigation is indeed characterized by a Dirac-like energy spectrum [14, 15].

It is shown that nonlinear transport experiments can be utilized to probe the

dispersion of two dimensional Dirac fermions.

Moreover, oscillations in the terahertz induced cyclotron resonance assisted

photocurrent and photoresistance are observed for all types of band struc-

ture, namely for normal and inverted parabolic, as well as linear [20]. These

oscillations are induced by the variation of an external magnetic field. The

corresponding experimental observations are presented in Chapter 6. It will

be demonstrated that they originate from the crossing of the Fermi level with

1 INTRODUCTION 7

Landau levels. This consecutive crossing results in the oscillations of spin

polarization and electron mobilities in the spin subbands (cones), which are

related to the de Haas-van Alphen and Shubnikov-de Haas effect.

Finally, Chapter 7 gives a conclusion of the main outcome of this work.

2 Theoretical Basics

This thesis is devoted to mercury telluride (HgTe), in particular to effects

concerning the special band structure of the material. Therefore, it is crucial to

understand the main characteristics of the dispersion and its dependency on the

width of the HgTe layer. These relations will be introduced in the first section

of this chapter as a simplified model of band ordering. The HgTe samples are

investigated by means of high frequency nonlinear opto-electronic phenomena,

namely terahertz induced photogalvanic and photoconductivity effects. Hence,

the following section will also depict the fundamental model of photocurrent

generation in parabolic semiconductor quantum well structures which is based

on an asymmetric spin dependent distribution of a charge carrier ensemble

in momentum space. Additional effects induced by terahertz radiation are

photoconductivity and cyclotron resonance, which will be introduced in the

third and forth section of this chapter. For interpretation and analysis of the

experimental results given in Chapters 5 and 6 quantum transport phenomena

are needed. Therefore, the last section will illustrate the Shubnikov-de Haas

and de Haas-van Alphen effect.

2.1 Band Structure of Mercury Telluride

HgTe belongs to the group of II-VI semiconductors and crystallizes in zinc-

blende structure [21]. In quantum well (QW) structures, sandwiched between

layers of cadmium telluride (CdTe), HgTe features an unique and very inter-

esting dispersion, which can be influenced in an extraordinary way. While the

conduction band of bulk CdTe is formed by s-states located at the group II

atoms and the valence band by p-states at the group VI atoms, the situation

for bulk HgTe is vice versa, yielding an inverted band ordering around the Γ

point of the Brillouin zone [12, 22]. This effect of an inverted band structure

is induced by the strong spin orbit interaction in this material [22, 23] and is

depicted in Fig. 1. For bulk CdTe the Γ6 band is s-type and lays energeti-

cally above the p-type Γ8 bands. These present conduction and valence band,

respectively. In case of bulk HgTe the picture is quite different: the p-type

Γ8 bands and the s-type Γ6 band are inverted compared to bulk CdTe. The

2 THEORETICAL BASICS 9

Figure 1: Energy spectrum of (a) bulk CdTe with a normal band ordering

and band gap Egap, as well as of (b) bulk HgTe with an inverted band

ordering and zero band gap between valence and conduction band, H1 and

LH1, respectively. The band ordering for CdTe/HgTe QWs is depicted in

(c) and (d): for QW widths LW smaller than the critical width LC the

electronic structure is normal, whereas in case of LW > LC it is inverted.

Picture according to Ref. [11].

conduction and valence band consist of the light hole band LH1 and the heavy

hole band H1 of the Γ8 bands and yield a zero gap system [18].

In HgTe based heterostructures this band ordering changes compared to the

HgTe bulk material. Due to confinement effects LH1 and H1 are separated by

an energy gap and conduction and valence band are now formed of the heavy


hole H1 and the electron E1 band [22]. According to Ref. [11] only these two

spin degenerated bands E1± and H1± are relevant, as they are close to the

Fermi energy.

The corresponding effective Hamiltonian near the Γ point is given by

Heff (kx, ky) =

(

H(k) 0

0 H∗(−k)

)

H = ǫ(k) + di(k)σi,

(1)

where σi are the Pauli matrices and

d1 +id2 = A(kx + iky),

d3 = M −B(k2x + k2

y),

ǫk = C −D(k2x + k2

y).

(2)

Here kx and ky are momenta in the plane of the two dimensional electron gas

and A,B,C and D are material specific constants. M is the Dirac mass pa-

rameter, indicating the order of bands. In conventional semiconductors e.g.,

CdTe, this parameter M is positive (E1 lays over H1). However, for HgTe hav-

ing an inverted band structure (H1 lays over E1) this parameter is negative.

Hence, in HgTe/CdTe QW structures the Dirac mass parameter M can be

tuned from positive to negative values simply by varying the QW width LW .

At a special QW width LC the mass parameter M is zero and a topological

quantum phase transition occurs, where the system merges from normal to in-

verted band ordering. At this point the structure is characterized by the Dirac

equation for massless particles [12]. This dependency of E1 and H1 on the QW

width LW is shown in Fig. 2 (a). Figure 1 (c) and (d) show the positions of

the two bands (H1 and E1) in a CdTe/HgTe QW for LW smaller and larger

than the critical width LC . In the former case the band ordering of CdTe dom-

inates, consequently the band structure is normal, whereas in the latter the

bands have the ordering of HgTe, namely an inverted band structure. Note,

that the band ordering can additionally be tuned by temperature and strain

for bulk and two dimensional HgTe structures, as well as by the fraction x in

Hg1−xCdxTe crystals, for review see Ref. [22, 24–26]. At the touching point of


the bands E1 and H1, indicated in Fig. 2 (a), the electronic structure of the

system is formed by a mixture of the two bands and shows a linear dispersion.

This so-called Dirac cone is depicted in Fig. 2 (b), where the colors indicate

the mixing of bands. A red shading stands for a dominant H1 state, whereas

in blue regions the dominant state is E1. Purple shading is a mixture of both.

According to Refs. [11, 15] the Dirac cone is formed by the four states

|E1,+1/2〉= f1(z) |Γ6,+1/2〉+ f4(z) |Γ8,+1/2〉 ,|H1,+3/2〉= f3(z) |Γ8,+3/2〉 ,|E1,−1/2〉= f1(z) |Γ6,−1/2〉+ f4(z) |Γ8,−1/2〉 ,|H1,−3/2〉= f3(z) |Γ8,−3/2〉 ,

(3)

where f1(z), f3(z) and f4(z) are the envelope functions, z is the growth direc-

tion, |Γ6,±1/2〉, |Γ8,±1/2〉 and |Γ8,±3/2〉 are the basis functions of the Γ6

and Γ8 band. These are degenerated for k = 0, and are coupled by an effective

Figure 2: (a) Energy of the E1 and the H1 band depending on the QW

width LW with the crossing point of bands at LW = LC . At this point the

band structure is characterized by a linear behavior, where two dimensional

massless Dirac fermions are predicted to occur. (b) shows the Dirac cone

for the critical width LC . Colors indicate the mixing of bands: while red

stands for a dominant H1 state, blue corresponds to the E1 state. Shades in

between (purple) indicate a mixture of both. Picture according to Ref. [11].


Hamiltonian at k 6= 0. Consequently, the spin-up and -down bands are formed

by a mixture of the different states of Eq. (3).

This Dirac-like dispersion can be achieved in different HgTe based systems.

The main subject of Chapter 5 is this linear band structure which has its ori-

gin in the critical QW width LC . Another possibility is provided by quantum

phase transitions at the interface of HgTe to materials with normal band struc-

ture. In case of a two dimensional system a Dirac-like dispersion arises at the

edges, whereas for three dimensional systems it occurs at the surfaces of HgTe.

These systems with edge and surface states belong to the class of topological

insulators, where dissipationless transport is possible by the mentioned states.

As topological insulators are not the subject of this thesis, their properties will

not be illustrated in more detail. For review see Refs. [10, 12].

2.2 Photogalvanic Effects

Terahertz (THz) radiation induced effects are very useful and efficient for the

investigation of semiconductor structures [17]. The photon energy in this spec-

tral range is much smaller than the energy gap of conventional semiconduc-

tors. Hence, no free carriers are generated across the band gap by absorption

of photons. As a consequence, effects based on the redistribution of carriers

in momentum space are observable. This redistribution results in a dc electric

current without external bias [17]. An example are THz radiation induced

photocurrents. Phenomenologically it is convenient to describe such dc cur-

rents by writing the coordinate and time dependent electric current density

j(r, t) = σE(r, t) as an expansion of powers of the electric field

E(r, t) = E(ω, q)e−iωt+iqr +E∗(ω, q)eiωt−iqr, (4)

at radiation frequency ω and photon wavevector q as [27]:

jα(r, t)=[

σ(1)αβEβ(ω, q)e

−iωt+iqr + c.c.]

+[

σ(2′)αβγEβ(ω, q)Eγ(ω, q)e

−2iωt+2iqr + c.c.]

+ σ(2)αβγEβ(ω, q)E

∗

γ(ω, q) + ....

(5)


Herein, Greek indices are Cartesian coordinates and c.c. stands for the com-

plex conjugate. Eq. (5) is limited to second order effects only. While the

first term on the right-hand side describes the linear transport, the second

and third term refer to second order electric field effects: the former term is

responsible for second harmonic generation, the latter one for time indepen-

dent contributions, which yield dc currents in response to the radiation’s ac

electric field. The first two terms are out of scope of this thesis which deals

with nonlinear transport phenomena proportional to the square of the electric

field (∝ Eβ(ω, q)E∗

γ(ω, q)). The lowest order of non vanishing terms for the

current density j relevant for the experimental observations of this thesis is

given by [17]

jλ =∑

µ,ν

χλµνEµE∗

ν , (6)

where E∗

ν = E∗

ν(ω) = Eν(−ω) is the complex conjugate of Eν and χ is a third

rank tensors. Equation (6) corresponds to the photogalvanic effect (PGE),

whose sum on the right-hand side can be also written with EµE∗

ν as a sum of

symmetric and antisymmetric product

EµE∗

ν = EµE∗

ν+ [EµE∗

ν ], (7)

yielding the photocurrent density to

jλ =∑

µ,ν

χλµν EµE∗

ν+∑

µ,ν

χλµν [EµE∗

ν ]. (8)

The photogalvanic effect is decomposed into the linear and circular photogal-

vanic effect (LPGE and CPGE), given by the first and second sum on the

right-hand side.

By applying an external magnetic field B, all the characteristic features of

the LPGE and CPGE change and new phenomena arise e.g., the magnetogy-

rotropic photogalvanic effect (MPGE). This effect will be introduced in the

following, as it is the relevant mechanism of current generation for almost all

findings presented in the experimental Chapters 4 to 6 of this thesis. It will

be discussed briefly with the help of a simplified phenomenological description

and then on the basis of a microscopic model.


Within the linear approximation of the magnetic field strength B, the current

density for the magnetic field induced photocurrent is given by [28,29]

jα =∑

β,γ,δ

φαβγδBβ EγE∗

δ+∑

β,γ

µαβγBβ eγE2Pcirc. (9)

Here, the fourth rank pseudotensor φ is symmetric in the last two indices and

µ is a third rank tensor. The first sum on the right-hand side is induced

by linear components of radiation and even by unpolarized light, whereas the

second sum requires circularly polarized radiation. This effect only occurs in

gyrotropic media which are characterized by the lack of inversion symmetry and

where nonzero components of third rank tensors exist [17, 30, 31]. Gyrotropic

media have second rank pseudotensors as invariants. This means that the

gyrotropic point group symmetry does not distinguish between polar vectors

e.g., current or electron momentum and axial vectors e.g., magnetic field or

spin [30]. This is the case for quantum well structures based on zinc-blende

semiconductors e.g., HgTe QWs.

Besides this phenomenological description, the MPGE can also be explained by

a microscopic model. Starting point is an electron gas homogeneously heated

up by Drude absorption of THz radiation. In gyrotropic media the electron-

phonon interaction contains an additional, asymmetric spin dependent term

and is given by [5]

Vkk′ = V0 +∑

α,β

Vαβσα(kβ + k′

β), (10)

where V0 describes the conventional spin independent scattering. V is a second

rank pseudotensor, σα represents the Pauli matrices and k (k′) is the initial

(scattered) wavevector. Note, that microscopically this additional term which

is proportional to σα(kβ + k′

β) has the same origin as the structure and bulk

inversion asymmetry [6,32–34]. This asymmetry in the electron-phonon inter-

action results in nonequal relaxation rates for positive and negative wavevectors

within one energy subband. This is depicted in Fig. 3 (a) and (b) for the spin-

down and -up subband. The difference in the arrows’ thickness corresponds

to the probability Wkk′ ∝ |Vkk′ |2 of electron-phonon scattering. This scatter-

ing probability is unequal for positive and negative k within one subband and


leads to an asymmetric carrier distribution in k-space. Each subband yields

an electron flux j±, where + and - stand for spin-up and -down. This effect

is called zero bias spin separation [5]. The two electron fluxes are of equal

strength but oppositely directed (j+ = −j−). Thus, the total electric current

is zero, but a pure spin current js is created given by [5]

js =1

2(j− − j+). (11)

By applying an external magnetic field B the two subbands split up energeti-

cally due to the Zeeman effect. The difference in energy is given by [35]

∆EZ = gµBB, (12)

where g is the effective Lande factor of the material and µB is the Bohr mag-

neton. As a result one of the subbands is now preferentially occupied. This

results in an imbalance of the two electron fluxes and therefore, they do not

cancel each other out anymore (see Fig. 3 (c)). This gives rise to a net electric

Figure 3: Asymmetric relaxation process of a homogeneously heated two

dimensional electron gas for (a) spin-down and (b) spin-up subbands. This

effect is called the zero bias spin separation. The difference in the arrows’

thickness corresponds to the scattering probability of electrons by phonons,

which is unequal for positive and negative k within one subband. This

yields an asymmetric carrier distribution and consequently an electron flux

j± in each subband. By applying an external magnetic field the subbands

split up energetically due to the Zeeman effect. This results in an imbalance

between the electron fluxes j+ and j− and is depicted in (c).


current jZ which is proportional to the Zeeman splitting ∆EZ and consequently

linear in B [29]:

jZ = −e∆EZ

EF

js = 4eSjs. (13)

Equation (13) is valid for a degenerated two dimensional electron gas and for a

low degree of spin polarization, namely for ∆EZ/2 << EF , with Fermi energy

EF [5, 28]. S is the magnitude of the average spin along the direction of B

and is given by S = n+−n−

2(n++n−), with carrier density n± in spin-up and -down

subband. For a degenerated two dimensional system the average spin is given

by S = −∆EZ

4EF[5].

Note, that there is a second microscopic mechanism contributing to the MPGE

which is related to the spin dependent indirect optical excitation of carriers

due to THz absorption in the presence of an external magnetic field. This

effect can be explained analogue to the asymmetric relaxation process described

above [5, 17].

2.3 Photoconductivity Effect

Besides THz radiation induced photocurrents, the radiation also results in a

change of conductivity of the semiconductor. For radiation in the THz range

and neglecting effects of impurity ionization, this phenomenon is limited to one

band as the photon energy is too small for the generation of free carriers across

the band gap. Hence, the carrier density n can be assumed to be constant

and the photoconductivity ∆σ is solely given by the change of mobility ∆µ

according to [17,35]

∆σ = |e|n∆µ, (14)

with elementary charge e and with respect to the dark conductivity without

radiation σ0 = |e|nµ0. µ0 is the mobility in the dark i.e., without THz radi-

ation. The change of mobility ∆µ originates from the change of the carriers’

temperature due to electron gas heating by THz absorption. This absorption

is drastically enhanced under cyclotron resonance conditions (see Section 2.4),


causing an intensified two dimensional electron gas heating. Therefore, mea-

surements of the photoconductivity represent one of the established methods

(next to radiation transmittance, see Section 3.3.4) of determining the effect

of cyclotron resonance [14, 36].

2.4 Cyclotron Resonance

Free carriers moving in a two dimensional layer with an external magnetic

field applied perpendicular to the QW plane experience a deflection due to

the Lorentz force FL = q[v ×B]. The carriers are bound to circular orbits in

k-space normal to the magnetic field with an angular frequency given by

ωc =qB

m∗. (15)

Herein, m∗ is the effective electron (hole) mass and q is the charge, which is

equal to -e for electrons and +e for holes [37]. High frequency radiation of

exactly this angular frequency ωc results in a resonant absorption, namely the

effect of cyclotron resonance (CR). The power absorbed by the free carriers for

circularly polarized radiation is given by [35]

P±(ω) ∝ 1

1 + (ω ± ωc)2τ 2, (16)

where the signs stand for right- (σ+) and left-handed (σ−) circular polarization

and τ is the momentum relaxation time. For linearly polarized light, which

is a superposition of σ+ and σ− polarization, the absorbed power adds up

to [35,37]

P linear(ω) ∝ 1

1 + (ω + ωc)2τ 2+

1

1 + (ω − ωc)2τ 2. (17)

A necessary requirement in order to observe CR is ωcτ ≫ 1. This condition

guarantees that the free carriers can perform a whole orbit before being scat-

tered by e.g., impurities. In quantum mechanics, for kBT < ~ωc, the effect

of CR can be considered as a resonant transition between successive Landau

levels (see Section 2.5) [35].

When the resonance condition is fulfilled (ω = ωc) and ωcτ ≫ 1 two cases have

to be distinguished: electrons and holes differ in their direction of rotation for


a fixed magnetic field polarity. Hence, electrons and holes only absorb one

kind of circularly polarized radiation [38].

Considering holes and positive magnetic fields (q = +e > 0, B > 0, conse-

quently ωc > 0 according to Eq. (15)) the resonant absorption of light is only

given for σ− polarization. The situation for electrons is vice versa. In the case

of linearly polarized light, the effect of cyclotron resonance occurs for both

carrier types and magnetic field polarities.

2.5 Quantum Transport Phenomena

The application of high magnetic fields perpendicular to a two dimensional

quantum well plane at low temperatures (kBT < ~ωc [39]) leads to a quanti-

zation of the energy spectrum into discrete Landau levels. In that case each

Landau level corresponds to a discrete δ peak in the density of states [39]. In

a realistic system these levels are broadened due to finite temperatures and

scattering. This is sketched in Fig. 4 (b).

The energy of the l-th Landau level for a parabolic energy spectrum is given

by [6, 40]

El = ~ωc(l +1

2) (18)

with l ∈ N0, yielding a distance between two neighbored Landau levels l and

l ± 1 of ∆El,l±1 = ~ωc. Taking into account Zeeman splitting according to

Eq. (12) the number of carriers on each spin split Landau level is given by

nLL = |e|B/h. For an electron gas with carrier density n the number of filled

Zeeman split Landau levels (so-called filling factor υ) is given by [40]

υ =n

nLL

=nh

|e|B, (19)

with Planck constant h. If n is fixed the Fermi energy EF oscillates as a

function of the applied magnetic field [40,41]. These oscillations i.e., the jumps

of EF depend on the density of states ν± [39]. If there are e.g., l Landau levels

completely populated and the (l + 1)-st is partly filled, thus the Fermi energy

EF lies within the level l + 1. If B increases the Landau level l + 1 shifts up


Figure 4: Sketch of (a) spin splitted Landau levels and Fermi energy EF as

a function of the magnetic field B for a parabolic dispersion and (b) density

of sates ν± for a fixed B = B1 with Landau level and Zeeman splitting,

∆El,l+1 and ∆EZ .

in energy and is emptied. Thus, EF falls back onto the level below [39]. This

is sketched for an ideal system with δ function Landau levels in Fig. 4 (a).

Herein, every jump of the Fermi energy corresponds to a filling factor υ: even

filling factors correspond to jumps between two Landau levels of different l,

whereas odd filling factors stand for jumps within a Zeeman split Landau level.

By variation of the magnetic field strength B, the Landau levels and corre-

sponding peaks in the density of states are pushed through the Fermi energy.

This consecutive crossing of El and EF can also be induced by keeping B fixed

and varying EF . Either results in an oscillating behavior of many physical

parameters which are periodic in 1/B. In this work oscillations of the mo-

bility of free carriers and their magnetic momenta are of interest, namely the

Shubnikov-de Haas and the de Haas-van Alphen effect.

Shubnikov-de Haas Effect Figure 4 (b) indicates that the density of states at

EF varies with the magnetic field B. If EF lies in between two Landau levels,

the density of states at EF is zero, whereas it has a maximum when EF is

in the middle of a broadened Landau level. This can be directly seen in a


system’s longitudinal resistance Rxx(B) which is illustrated in Fig. 5 (a) for a

8 nm HgCdTe QW Hall bar structure at 4.2 K. The resistance has a minimum

when EF lies between two Landau levels (zero density of states) and is finite

when it lies within a Landau level [39]. This oscillating behavior of Rxx as a

function of B is the Shubnikov-de Haas (SdH) effect. Furthermore, Fig. 5 (a)

demonstrates that for low magnetic fields Rxx(B) is constant. This is due to

the fact that when B goes to 0, the Landau levels decrease their distance in

such a way that they eventually overlap. In this case EF always lies within

this constant density of states and the resistance becomes constant [40].

Each minimum of Rxx(B) can be related to a filling factor υ (indicated in

the figure) which plotted against 1/B yields a straight line with a slope of

hn/e. This is shown in the inset of Fig. 5 (a). Hence, the Shubnikov-de

Haas oscillations are periodic in 1/B. The analysis of the oscillations can

provide information about the carrier density, occupation of higher subbands,

which can be seen by two different frequencies in the SdH oscillations, or

the evolution of Zeeman spin splitting [39]. The latter can be seen in Fig. 5

(a): for magnetic fields larger than ≈ 2.76 T (value corresponds to tiny dip

between filling factors 10 and 12) even and odd filling factors are observed.

This indicates the resolution of the Zeeman splitting of Landau levels.

De Haas-van Alphen Effect Another effect similar to the Shubnikov-de Haas

effect originates from a periodic change of the total energy of the free carriers

with the magnetic field, namely the de Haas-van Alphen effect. This can be

detected experimentally as oscillations of the material’s magnetic momentum

µm = −∂E/∂B [41]. For matters of simplification the spin is neglected in the

following. The variation of the total energy of free carriers is depicted in Fig. 5

(b).

For B = 0 T the density of states is continuous (regions I and IV). When a

magnetic field is applied the carriers squeeze onto discrete Landau levels. For

the magnetic field B1 depicted in region II the system has the same energy

as for zero magnetic field (region I): the number of carriers which raise their

energy is equal to the number of carriers which degrade their energy in order

to divide onto the Landau levels. When B is increased to a value of B2, the


total energy of the system increases as the highest carriers shift energetically

up (region III). For B3 the system’s total energy is back to the initial one (as in

I, II or IV) as the uppermost carriers have to decrease in energy (region V) [41].

Consequently, the total energy has a minimum at points such as B1, B3, ...

and maximum points at B2, .... This oscillation in energy with a periodicity of

1/B can be seen in the free carriers’ magnetic momentum and hence, in their

spin S, as µm ∝ S.

Figure 5: (a) Measurement of Shubnikov-de Haas oscillations in the longi-

tudinal resistance Rxx of a 8 nm HgCdTe Hall bar sample. Numbers indicate

filling factors υ. The inset shows the filling factor υ as a function of the in-

verse magnetic field 1/B. The de Haas-van Alphen effect is depicted in (b)

as a simplified model of the free carriers’ total energy; the Zeeman effect is

neglected. Whereas the density of states is continuous for B = 0 T (regions

I and IV), it splits into Landau levels when a magnetic field is applied. In

the case of B1 (II) and B3 (V) the total energy is the same as for B = 0 T

as the same amount of carrier decrease their energy, as increase it. For B2

the total energy is increased as the uppermost carriers shift energetically up

(III).

3 Experimental Methods

This chapter is devoted to the experimental methods used for the investigations

in this work. In order to give a clear picture of the performed measurements

the whole experimental setup is depicted first. Then the studied samples are

introduced, which can be divided into two groups: diluted magnetic semicon-

ductor (DMS) QW structures and - forming the core of this thesis - HgTe based

QW structures. In the third section of this chapter the different measurement

techniques are illustrated. These comprise mere electrical and optical, as well

as opto-electronic measurement techniques. As the variation of polarization of

the THz radiation plays an important role for some of the studied effects e.g.,

cyclotron resonance, two types of waveplates will be presented.

3.1 Experimental Setup

The experimental setup used for the measurements includes a laser system, an

optical cryostat with a superconducting split coil magnet, as well as optical

chopper systems and other devices. They are all combined in a complex struc-

ture, which will be introduced in the following. The whole experimental setup

can be seen in Fig. 6.

The laser system consists of an electrically pumped continuous wave (cw) PL5 1

CO2 gas laser, emitting in the mid-infrared (MIR) range, and a 295FIR1 gas

laser. This is optically excited by the CO2 gas laser and emits in the far-

infrared range (FIR). The THz radiation of the FIR laser is line tunable and

its wavelength depends on the kind of gas filled into the cavity, as well as on the

exact wavelength of the exciting MIR radiation. For this thesis a wavelength

of 118.8 µm was used. The parameters of the operating laser system are given

in Tab. 1.

The CO2 gas laser belongs to the group of vibrational and rotational gas lasers

and is one of the most important representatives of the group of molecular

lasers. It is longitudinal electrically excited and delivers a power of approxi-

mately 35 W. The emitted wavelength can be tuned around and between the

1Edinburgh Instruments, Livingston, UK

3 EXPERIMENTAL METHODS 23

Figure 6: Experimental setup with MIR (CO2) and FIR gas laser system.

The optical path is sketched as a red line. Most devices are labeled in the

picture. 1: optical chopper, 2: waveplates or other optical devices, 3: planar

mirror, 4: parabolic mirror, 5: digital multimeter, 6: lock-in amplifier, 7:

Golay cell. A detailed description is given in the text. Picture modified

after Ref. [42].

lines of 9.4 and 10.4 µm. These lines belong to two different optical active

transitions between vibrational modes of the CO2 molecule. The line tun-

ing is possible as these vibrational modes split up into different rotational

modes [17, 43]. The output radiation is not used for experiments, but acts

only as an optical pump source for the second molecular gas laser. Hence,

the radiation is deflected by two planar mirrors and focused by a zinc selenide

(ZnSe) lens through a polarizing ZnSe Brewster window mounted on top of

a steel cone into the cavity of the FIR laser system. The Brewster window

ensures linearly polarized light, with the E-field vector being aligned horizon-

tally. The incoming MIR radiation excites a vibrational state of the molecule

of the second gas laser system having a permanent electric dipole moment.


λ MIR laser (µm) FIR laser gas λ (µm) f (THz) E~ω (meV) P (mW)

9.695 methanol 118.8 2.53 10.35 60

Table 1: Parameters of the operating laser system: excitation wavelength

λ of the CO2 gas laser; laser gas of the FIR laser system; wavelength λ,

frequency f , photon energy E~ω and power P of the emitted THz radiation.

This is sketched in Fig. 7 (a) for a symmetrical top molecule, where K is the

projection of the angular momentum J on the symmetry axis of the molecule.

Assuming that the vibrational relaxation is slow enough, lasing transitions oc-

cur between rotational states only emitting radiation in the FIR range [17].

The cavity of the FIR laser is closed by a silver coated z-quartz window, which

is transparent for wavelengths with frequencies in the THz range, but reflects

the MIR radiation and so ensures a monochromatic output. The transversal

mode shape can be adjusted by variation of the resonator length via screws at

the silver coated quartz window. It can be monitored by thermal paper or by

a Pyrocam III2 in order to verify that the beam has a Gaussian shape (see

Fig. 7 (b)). The emitted monochromatic FIR radiation of the molecular gas

laser is directed through a beam splitter, consisting of Mylar3. The reflected

part of the beam (≈ 15%) is used as a reference signal in the analysis of the

measurements and allows to monitor the stability of the laser. It is modu-

lated by a 300CD4 optical chopper with a frequency of fref = 140 Hz and

directed to a pyroelectric reference detector LIE-329-Y 5 by a planar mirror.

This reference signal Uref is measured by a KI2000 6 digital multimeter and

sent to a control computer running a LabVIEW7 program. This LabVIEW

program controls the whole experimental setup and records all measured data.

It was written by Wolfgang Weber8 and Sergey Danilov8. The main part of

the beam (≈ 85%) is transmitted through the beam splitter and is modulated

by a second 300CD4 optical chopper at a frequency of fsig = 620 Hz. At this

2pyrometer for temperature detection, Spiricon Inc., Logan, UT, USA3a polyester film made from stretched polyethylene terephthalate4Scitec Instruments, Wiltshire, UK5Laser Components, Olching, Germany6Keithley Instruments, Germering, Germany7Laboratory Virtual Instrumentation Engineering Workbench8Terahertz Center, Regensburg, Germany


point additional devices e.g., waveplates or analyzers can be inserted into the

optical path. The integration of a step motor, which is operated by LabVIEW

and rotates the mounted waveplates step wise, is possible. The beam is then

deflected by a planar mirror and is focused by a parabolic9 one into a Spec-

tromag SM4000-8 10 optical cryostat. The inner windows of this cryostat are

made of z-cut quartz, the outer windows are fabricated of TPX11. As these

are transparent for THz and visible light, the latter are covered with a black

polyethylene sheet to prevent illumination of the sample with ambient light.

The sample is located in the middle of the cryostat between two supercon-

ducting split coil magnets, allowing magnetic fields up to 7 T perpendicular to

the sample’s surface. The magnetic field can by controlled by an IPS120-10 10

9focal length f = 22.5 cm, Kugler GmbH, Salem, Germany10Oxford Instruments, Abingdon, UK11polymethylpenten

Figure 7: (a) Scheme of optically pumped THz transitions in a symmetric

top molecule. Electrons are excited to a higher vibrational mode by the

incoming CO2 gas laser radiation, yielding a population inversion in both

states. Relaxation transitions occur between the rotational modes of both

vibrational states, emitting radiation in the THz range. (b) Spatial beam

shape with an average power of ≈ 60 mW, recorded with the Pyrocam III 2.


power supply. The temperature at sample position can be varied from liquid

helium temperature (T = 4.2 K) up to room temperature, is maintained and

measured by a LS332 12 temperature controller. A wiring system connects the

sample with two SR830 DSP13 lock-in amplifiers, with an input resistance of

10 MΩ, allowing the measurement of two different voltage drops across the

sample (compare Section 3.2 and 3.3). A part of the radiation is transmitted

through the sample and leaves the cryostat on the other side. It is then par-

allelized and focused, both by a parabolic mirror14, to a Golay cell detector15,

whose output can be recorded via lock-in technique. The lock-ins are con-

nected to the control computer via GPIB16 as well as the reference detector

by a digital multimeter, temperature and magnetic field controller and step

motor unit. As mentioned before, all variables are recorded and can be set by

the LabVIEW program.

3.2 Investigated Samples

In the framework of this thesis two different groups of semiconductor struc-

tures were investigated: HgTe based quantum well heterostructures of similar

layer design, but different QW widths LW and two diluted magnetic semicon-

ductor QW structures. Their designs and properties will be introduced in the

following.

3.2.1 Mercury Telluride Quantum Well Samples

The main part of this thesis is devoted to photocurrents in HgTe based QW

structures. The samples were provided by Nikolai N. Mikhailov17 and Sergey

A. Dvoretsky17. They were grown by molecular beam epitaxy (MBE) onto a

(013) oriented gallium arsenide (GaAs) substrate [44]. These (013) grown QWs

belong to the C1 point group possessing no symmetry restriction. The HgTe

12Lake Shore Cryotronics, Westerville, OH, USA13Stanford Research Systems, Sunnyvale, CA, USA14focal length f = 22.5 cm, Kugler GmbH, Salem, Germany15Artas GmbH, Zeitlarn, Germany16General Purpose Interface Bus17Institute of Semiconductor Physics, Novosibirsk, Russia


Figure 8: (a) shows a cross section of the layer design of the investigated

HgTe based QW samples. The QW layer with width LW is indicated in red.

(b) depicts a typical result of a magneto-transport measurement by which

all samples were characterized. The results of a 6.6 nm wide HgTe QW are

presented for two different carrier types and densities p1 and n2 (see Chap-

ter 5). The upper panel corresponds to a p-type sample, while the lower one

shows a n-type behavior. Solid lines show the Hall resistance Rxy with Hall

plateaus, dashed lines are the longitudinal resistance Rxx with Shubnikov-de

Haas oscillations. They scale on the right and left, respectively.

QW is sandwiched between two layers of Cd0.7Hg0.3Te. Figure 8 (a) shows an

exemplary cross section. Depending on the QW width LW the band structure

can change its ordering, as described in Section 2.1. This makes it possible to

divide the samples into three groups of band structure: normal and inverted

parabolic as well as linear.

The samples were prepared in different designs e.g., large area square shaped

samples and Hall bars (see Fig. 9). These various designs are due to the differ-

ent kind of measurements performed on the samples. While Hall bar structures


width LW (nm) design gating band structure

5 square no normal

6.4 Hall bar no linear

6.6 Hall bar yes & no linear

6.6 square no linear

8 Hall bar∗ no normal

8 square no inverted

21 square no inverted

Table 2: QW width LW , design, gating and band structure of the investi-

gated HgTe QW samples. ∗The QW contains cadmium: Hg0.86Cd0.14Te.

show best resolved Shubnikov-De Haas oscillations and quantum Hall plateaus,

they are not ideal for photocurrent or transmission measurements. For this

purpose the samples have to be large enough to ensure no illumination of the

contacts or samples’ edges. Both can both give rise to other signals superim-

posing the investigated effects. Therefore, radiation transmittance, as well as

photocurrent experiments are mainly studied on large area 5×5 mm2 square

samples, where the indium (In) contacting was done at the samples’ edges and

corners only (see next section).

An overview of the studied HgTe QW samples, their design and band struc-

ture is given in Tab. 2. All the samples have been characterized by magneto-

transport measurements, as described in Section 3.3.1. Carrier densities are

given in Chapters 5-6, where the experimental results are presented.

Normal Parabolic Band Structure Two samples with normal band structure

were investigated. The first one is a pure HgTe QW with a width LW = 5 nm,

which is smaller than the critical width LC . Hence, it is characterized by a

normal parabolic band ordering where the E1 band lies above the H1 band

as known from conventional semiconductors (e.g., GaAs, CdTe or InAs). The

sample’s design is a large area square with an edge length of 5 mm. Eight

ohmic In contacts are soldered to the corners and to the middle of the edges

(compare Fig. 9 (a) and (b)). These ohmic contacts are connected to the pins


of an eight-pin chip carrier via gold wires, which allow the measurement of the

voltage drops induced by bias or THz radiation.

The second sample is a 8 nm HgCdTe QW Hall bar structure. It has a 14% Cd

content in the QW, in contrast to all other pure HgTe QWs. As a consequence,

the transition from normal to inverted parabolic band structure is obtained for

wider QWs than for pure HgTe QWs [26, 45]. In case of the here investigated

Hg0.86Cd0.14Te sample with a width of LW = 8 nm the dispersion is normal

parabolic. The Hall bar structure is depicted in Fig. 9 (c) and (d). It is 50 µm

wide, with distances between the contacts of 100 and 250 µm. The contact

pads of the Hall bar are soldered in the same way as the contacts of the square

sample i.e., via In contacts and gold wires. The sample is mounted on an

eight-pin chip carrier.

Linear Band Structure Increasing the QW width LW the E1 and H1 bands

touch each other at a critical width LC of approximately 6-7 nm (see Fig. 2

(a)) [11,13,46]. At this point the dispersion shows a linear, Dirac-like behavior.

For this thesis a quantum well width of 6.6 nm was selected. Samples in both

shapes, Hall bar and square, were investigated. Two Hall bar structures were

available, prepared with and without gate. A gate enables a controllable way

of carrier density variation. Therefore, an isolating layer of 100 nm silicon

dioxide (SiO2) and 200 µm silicon nitride (Si3N4) is used with a metallic gate

layer of titanium gold (TiAu) deposited on top. The gate layer can be biased

via a pin of the eight-pin chip carrier to which it is connected by a gold wire.

Inverted Parabolic Band Structure As a last group samples with a QW width

LW larger than the critical width LC were under study. For this case the elec-

tron band E1 and heavy hole bandH1 are exchanged in contrast to conventional

semiconductors e.g., CdTe. Hence, they are characterized by an inverted band

ordering. Two samples were available, with widths of 8 and 21 nm. The sam-

ples’ design is a 5×5 mm2 square shape. The contacting is done in the same

way as described above.


Figure 9: (a) shows a picture of a square shaped sample mounted on a

chip carrier. It is connected to the pins via gold wires and In soldering.

(b) gives a schematic drawing of (a). (c) depicts a Hall bar structure with

a width of 50 µm and distances between contacts of 100 and 250 µm. (d)

shows the wiring of the Hall bar sample on a chip carrier.

3.2.2 Diluted Magnetic Semiconductor Samples

In DMS structures paramagnetic ions e.g., manganese (Mn2+), are introduced

into the host material. These Mn2+ ions provide magnetic moments, which

give the opportunity of a controllable variation of the magnetic properties of

the semiconductor. This can be done by e.g., using different host materials,

changing the Mn2+ concentration or varying the temperature. In the following,

two DMS samples are investigated with different host materials: a CdTe based

and an indium arsenide (InAs) based QW structure. Both samples are (001)

MBE grown QW structures on a GaAs substrate. They are fabricated as

large area squares of the same dimensions and contact preparation as the

HgTe samples (see Section 3.2.1). Their properties are summarized in Tab. 3

of Section 4.1. (001) grown CdTe and InAs QW structures belong to the C2v

symmetry group. Both have two mirror planes perpendicular to the QW and a

C2 axis parallel to the growth direction. Depending on the host material (CdTe,

II-VI or InAs, III-V) the Mn2+ ions result in different electrical properties.


Among the DMS materials Cd(Mn)Te is one of the most intensely studied semi-

conductor heterostructures. The sample was provided by the group of Tomasz

Wojtowicz18. The 10 nm wide Cd(Mn)Te QW is embedded between two layers

of Cd0.7Mg0.3Te (see Fig. 10 (a)). During the growth process three monolay-

ers of Cd0.86Mn0.14Te were inserted applying digital alloy technique [47–49].

CdTe belongs to the group of II-VI semiconductors. In this case the Mn2+ is

electrically neutral as it substitutes Cd2+, which has the same valence. But

it provides a localized spin of 5/2 [18]. The sample is iodine (I) doped and

characterized by a n-conductivity.

The second DMS sample has an InAs host material. InAs is, in contrast to

CdTe, a narrow band gap material just like HgTe. The structure is MBE grown

by Dieter Schuh19 and Ursula Wurstbauer19 and has a 4 nm wide In(Mn)As

QW embedded in a 20 nm In0.75Ga0.25As channel. Fig. 10 (b) depicts a cross

section of the structure with a Mn2+ doping layer grown before the QW. Due

to segregation processes the QW features a limited Mn2+ concentration [50–

52]. InAs belongs to the III-V semiconductor group. Therefore, the Mn2+

18Institute of Physics, Warsaw, Poland19Terahertz Center, Regensburg, Germany

Figure 10: (a) depicts a schematic cross section of the Cd(Mn)Te sample.

Herein, three monolayers of Cd0.86Mn0.14Te were inserted during the growth

of the 10 nm wide QW. (b) shows the In(Mn)As structure which has a Mn2+

doping layer inserted before the QW. Due to segregation of the manganese

during the growth process, the 4 nm wide QW has a limited concentration of

Mn2+. In (c) the experimental geometry is sketched, with inplane magnetic

field and normal incidence of radiation.


ions provide free holes in addition to the localized magnetic moments as they

substitute In3+. This results in a p-type conductivity of the In(Mn)As DMS

sample [50, 53].

3.3 Electrical and Optical Measurement Techniques

At this point the different measurement techniques, which were used in this

work, will be introduced. These comprise magneto-transport measurements,

analysis of THz radiation induced photocurrents and photoconductivity (pho-

toresistance), as well as the detection of the transmitted radiation through the

sample.

3.3.1 Characterization via Magneto-Transport Measurements

Standard magneto-transport measurements were performed for every sample

in order to determine carrier type and density. For this kind of measurement

the sample was ac biased with 1 V. The current was limited by a 1 MΩ resistor

giving an ac current of 1 µA through the sample. An external magnetic field

was applied normal to the QW plane. The signals Ux and Uy were measured

with lock-in technique via a voltage drop over a 10 MΩ load resistor (compare

Fig. 11 (a)). These signals correspond to the Hall and longitudinal resistance,

respectively, using R = U/Iac. In most cases a Van-der-Pauw geometry was

used. An exemplary result of such a measurement, which was performed on

a 6.6 nm HgTe QW sample (see Chap. 5), is depicted in Fig. 8 (b) for two

different carrier types and densities. Well pronounced Shubnikov-de Haas os-

cillations and quantum Hall plateaus can be seen in the longitudinal (Rxx) and

Hall (Rxy) resistance. The different slopes of the Hall resistance indicate the

different carrier types: holes in the upper panel and electrons in the lower one.

To achieve a controllable variation of the carrier density some samples were

prepared with a gate (compare Section 3.2) or the effect of persistent photo-

conductivity was used [14, 54]: by illuminating the sample with a red light

emitting diode (λ = 630 nm) for a defined time ti the carrier density can be

changed. This process can be undone by heating the samples up to ≈ 150 K.

In the following this procedure is referred to as optical doping [54, 55].


Figure 11: Overview of the different measurement setups with a magnetic

field B applied normal to the sample’s surface for (a) magneto-transport,

(b) photoconductivity (photoresistance), (c) photocurrent and (d) radiation

transmission. In (a) an ac current is sent through the sample and two voltage

drops Ux and Uy (the latter is not shown) are measured. These signals are

proportional to the Hall and longitudinal resistance. The setup for (b) is

similar but with a dc current and THz radiation of normal incidence. For

(c) the sample is unbiased but also with THz radiation. For (a)-(c) the

signals Ui, with i = x, y, are measured as a voltage drop over a 10 MΩ load

resistor via lock-in technique. Note, that the y direction can be measured

analog to the x direction which is indicated in the figure. (d) presents a pure

optical measurement, where the transmitted radiation through the sample

is detected by a Golay cell detector.


3.3.2 THz Radiation Induced Photocurrents

For the measurement of photocurrents the unbiased sample was illuminated

under normal incidence with THz radiation. An external magnetic field was

applied perpendicular to the sample’s surface. This radiation induces photo-

currents, which were measured analog to Section 3.3.1, via a voltage drop over

a 10 MΩ load resistor applying lock-in technique. The corresponding setup is

depicted in Fig. 11 (c). In order to eliminate fluctuations of the laser power

during a measurement the signals Ux and Uy were normalized by the recorded

reference signal Uref . Then they were multiplied with the power calibration

factor Pcalib given in units of (V/W) according to (Ui × Pcalib)/Uref in (V/W)

with i = x, y. The calibration factor was obtained with a Vector H410 20 pow-

ermeter by measuring the averaged laser output power P and normalizing it

to a mean value of reference signal Uref .

Note, that for the DMS samples, with C2v symmetry, photocurrents are for-

bidden for normal incidence of light and a perpendicular magnetic field [24].

Therefore, the study of THz radiation induced photocurrents was done with

an inplane magnetic field. This is shown in Fig. 10 (c).

3.3.3 Measurement of Photoresistance

As a complementary measurement studies of the photoresistance (or photocon-

ductivity) ∆R were performed. ∆R corresponds to the change of the sample’s

resistance when illuminating the structure with THz radiation. In order to dis-

tinguish between the two THz radiation induced effects of photocurrent and

photoresistance, the sample was biased individually either with a positive or a

negative dc electric current of |Idc| = 1 µA (±1 V, 1 MΩ). The magnetic field

B and the propagation direction of the THz radiation were aligned normal to

the sample’s surface. The voltage drops Ui,±, with i = x, y, for both currents

±Idc were recorded as sketched in Fig. 11 (b). While the effect of photoresis-

tance (photoconductivity, UPCi,± ) is dependent on the current direction through

20Scientech Inc., Boulder, CO, USA


the sample, the photogalvanic effect (UPGEi,± ) is not effected. Consequently, the

difference of these two signals divided by the current, given by

∆Ri =Ui,+ − Ui,−

2Idc=

(UPCi,+ + UPGE

i,+ )− (−UPCi,− + UPGE

i,− )

2Idc, (20)

corresponds to the change of resistance. It is inverse proportional to the change

of conductivity and to the difference in mobility µ (see Section 2.3). These

measurements are performed in addition to opto-electronic photocurrent mea-

surements - the main topic of this work - and provide access to additional

information e.g., on cyclotron resonance [36].

3.3.4 Measurement of Radiation Transmittance

For a further understanding measurements of radiation transmission were per-

formed, which is a standard method for the identification of the effect of cy-

clotron resonance. The corresponding setup is shown in Fig. 11 (d) with the

magnetic field and the propagation direction of the THz radiation aligned nor-

mal to the QW plane. Radiation transmittance is a sheer optical measurement

technique; the sample was therefor without bias and no wiring was needed. A

Golay cell21 behind the sample detected the transmitted part of light and gave

the signal to a lock-in amplifier. As the Golay cell is a very slow detector the

modulation frequency fsig of the radiation was reduced to ftm ≈ 30 Hz. Due to

high thermal noise on the one hand and a slight hysteresis effect of the detec-

tor on the magnetic field on the other hand, every transmission measurement

was recorded twice (for both magnetic field sweeping directions) and averaged

afterwards.

21Artas GmbH, Zeitlarn, Germany


3.4 Development and Characterization of THz Wave-

plates

As the variation of the polarization state of radiation is important for many

aspects of this work, in particular for the cyclotron resonance assisted mea-

surements, it was necessary to use waveplates suitable for the THz range and

of best possible quality.

In cooperation with Tydex22 a special waveplate was developed applicable for

a wide spectral region in the THz range, which is called achromatic waveplate.

The analysis of the conversion of linearly to circularly polarized light was done

by rotating an analyzer mounted in a stepmotor unit behind the waveplate and

measuring the transmitted radiation intensity with a LIE-329-Y 23 pyroelectric

detector. Setup and results can be seen in Fig. 12 (a) and (c) for a wavelength

of λ = 118.8 µm. The transmitted radiation intensity is almost independent of

the angle β of the analyzer. This means that the Tydex achromatic waveplate

demonstrates a high quality of polarization conversion to circularly polarized

light. Similar results were obtained for other wavelengths with frequencies in

the THz range. Details can be seen in Ref. [56].

In addition a waveplate made for a wavelength of λ = 118.8 µm from QMC

Instruments24 was verified. It features a special anti-reflection coating which

increases the transmittance behavior of the z-cut quartz plates. As the plates

had no indication of the optical axis it had to be determined experimentally.

For this purpose the linearly polarized radiation passes through the waveplate

and then through an analyzer, aligned parallel or perpendicular to the initial

linear polarization. The transmitted radiation is then detected with a LIE-329-

Y 22 pyroelectric detector as a function of the angle ϕ′

. The angle ϕ′

of the

waveplate is varied in 2 steps from 0 to 360 perpendicular to the propagating

direction of the radiation. This setup can be seen in Fig. 12 (b). The signal of

the transmitted radiation is measured for both grid alignments. This is shown

in Fig. 12 (d). The two touching points at ϕ′ ≈ 67 and 157 correspond to

circularly polarized radiation. For this case the horizontal and vertical linearly

22St.Petersburg, Russia23Laser Components, Olching, Germany24West Sussex, UK


polarized part of light have the same intensity. In order to analyze the quality

of the waveplate the same measurement as for the achromatic waveplate was

done (see Fig. 12 (a)). The result is shown in Fig. 12 (e). With a deviation from

the mean signal of only 7% the waveplate demonstrates an excellent quality.

Many measurements of this thesis were carried out with right- or left-handed

circularly polarized radiation. In these cases the waveplate from QMC Instru-

ments was used.

Figure 12: (a) and (b) show the setup for the analysis of the waveplates’

quality and estimation of the optical axis. (c) depicts the transmitted laser

intensity in dependency of the analyzer rotation angle β for the achromatic

waveplate from Tydex for a wavelength of λ = 118.8 µm. In (d) the trans-

mitted radiation is detected depending on the rotation angle ϕ′

of the QMC

Instruments waveplate for two different grid positions of the analyzer and

λ = 118.8 µm. (e) shows the same as (c) but for the QMC Instruments

waveplate. The alignment of the waveplate is chosen as ϕ′

= 67 which

corresponds to the first touching point in (d), where the transmitted part

of radiation is fully circularly polarized.

4 Magnetogyrotropic Photogalvanic Effect in

a Diluted Magnetic Semiconductor Hetero-

structure

The magnetogyrotropic photogalvanic effect is a rather novel opto-electronic

method for the investigation of semiconductor heterostructures. In order to

understand its spin dependent origin a well established semiconductor system

will be analyzed before turning to the core of this thesis - the study of THz

radiation induced photocurrents in HgTe based QW systems. A suitable can-

didate for this are diluted magnetic semiconductor heterostructures as they are

well characterized and provide an opportunity to control the sample’s magnetic

properties. In this chapter the experimental observations for a Cd(Mn)Te and

In(Mn)As DMS sample will be presented and discussed by means of the MPGE

(compare Section 2.2). The results are published in Ref. [19]. Only a part of

the experimental results of Ref. [19] are presented here.

4.1 Experimental Results

The THz radiation induced photocurrents in the unbiased DMS samples were

measured as a function of a magnetic field By applied in the QW plane for

different fixed temperatures. In addition, measurements of the temperature

dependency of the photocurrents for a given magnetic field value were per-

formed. As the measured directions x and y show a similar behavior, only

the photocurrent jx normalized by the laser power P will be discussed here.

Furthermore, the observed currents have been shown to be an odd function

of the magnetic field. Therefore, the presented curves are limited to positive

By only. For excitation linearly polarized light was applied normal to the QW

plane with a wavelength of 118.8 µm.

Figure 13 (a) shows the THz radiation induced currents jx/P measured in the

Cd(Mn)Te sample for temperatures of 1.8, 4.2, 8 and 40 K as a function of the

inplane magnetic field. Note, that some curves are multiplied by a factor for a

better comparison, indicated near the corresponding curve. For high tempera-

tures or for low temperatures and small magnetic fields, the magnitude of the

4 MPGE IN A DMS HETEROSTRUCTURE 39

photocurrent shows a linear behavior (solid lines in Fig. 13 (a)). Whereas, for

low T and high By the current saturates with increasing magnetic field and for

T = 1.8 K even shows a decrease for magnetic fields larger than ≈ 3 T. The

maximum current of -0.74 µA/W is observed for T = 1.8 K and By = 3 T.

In general the amplitude of jx/P decreases drastically with increasing T . In

order to study this temperature dependency further, additional measurements

were performed at a constant inplane magnetic field of 2 T. The photocurrents

were measured while cooling down the sample slowly from 150 to 1.8 K. This

is depicted in Fig. 14 (a). The amplitude increases strongly by more than two

orders of magnitude in the investigated temperature range. The current has

a positive maximum of +7.4 nA/W at approximately 21.4 K, changes sign at

about 13 K (inset in Fig. 14 (a)) and exhibits an absolute maximum value of

-0.69 µA/W at 1.8 K.

Figure 13: Magnetic field dependency of the normalized photocurrent

jx/P for the (a) Cd(Mn)Te and (b) In(Mn)As QW sample at different tem-

peratures. The THz radiation was linearly polarized with a wavelength of

118.8 µm. Note, that some curves in (a) are multiplied by a constant factor

for a better comparison. The factors are indicated in the figure. Solid lines

are guides for the eye.


For the In(Mn)As sample the photocurrents as a function of the magnetic

field at constant temperatures (Fig. 13 (b)) and of the temperature at a fixed

magnetic field (Fig. 14 (b)) show similar behaviors as for the Cd(Mn)Te struc-

ture. The linear dependency on By for small magnetic fields is observed for

the whole temperature range and is indicated by solid lines in Fig. 13 (b). For

T = 1.8 K the current saturates at a value of approximately -64.0 nA/W for

fields larger than ≈ 3 T. Furthermore, the amplitude of the current drastically

increases when the sample is cooled down from 150 to 1.8 K. It has a maximum

amplitude of -55.2 nA/W at approximately 2 K and 2 T. In contrast to the

Cd(Mn)Te sample, however, there is no obvious change of sign in the whole

temperature range (see inset in Fig. 14 (b)).

Figure 14: Temperature dependency of the normalized photocurrent jx/P

at a fixed magnetic field value of By = 2 T. The incident radiation was

linearly polarized with a wavelength of 118.8 µm. Panel (a) and (b) corre-

spond to the Cd(Mn)Te and the In(Mn)As DMS QW sample. The insets

depict a zoom of the photocurrent in the temperature range of 10 to 150 K.

While the photocurrent of the Cd(Mn)Te sample shows a zero crossing at

approximately 12 K, no change of sign is detected for the In(Mn)As sample.


Similar results have been obtained for different wavelengths with frequencies

even ranging into the microwave spectrum. These findings are not presented

here but can be viewed in Refs. [19, 57, 58]. In addition, experimental ob-

servations on different DMS heterostructures are discussed in the mentioned

references. These structures comprise CdTe and InAs based DMS samples

varying e.g., in the manganese concentration or layer design and include n-

type In(Mn)As hybrid structures as well.

sample carrier density (cm−2) mobility (cm2/Vs) design

Cd(Mn)Te 6.2 × 1011 (n-type) 16000 square

In(Mn)As 4.4 × 1011 (p-type) 2000 square

effective g factor N0αe/h (eV) x

Cd(Mn)Te -1.64 0.22 0.013

In(Mn)As -15 -1 0.01

Table 3: Parameters of the two investigated DMS samples. Carrier density

and mobility are obtained at 4.2 K without illumination. Effective g factor

and exchange integral N0αe/h for electrons (e) and holes (h) are obtained

from Ref. [18] and Refs. [59, 60] for Cd(Mn)Te and In(Mn)As, respectively.

x represents the effective manganese concentration in the QW [19,50].

4.2 Discussion

The experimental results of the THz radiation induced photocurrents in DMS

structures can be well explained by the magnetogyrotropic photogalvanic ef-

fect. In fact, they provide a direct experimental proof for the spin related

mechanism based on the zero bias spin separation (see Section 2.2). Spin po-

larized current generation is possible by illuminating an unbiased QW structure

with THz radiation. This results in a homogeneous heating of the electron gas.

The relaxation of carriers is asymmetric in k-space and yields an electron flux

in each spin subband j+ (spin-up) and j− (spin-down) of equal strength but

oppositely directed. For zero magnetic field these fluxes form a pure spin cur-

rent according to Eq. (11). By the application of an external magnetic field

these two fluxes get unbalanced and result in a measurable net electric current


jZ , which is proportional to the Zeeman splitting (see Eq. (13)). This propor-

tionality was the reason to choose DMS structures in order to establish the

spin dependent features of the MPGE as the Zeeman splitting can be varied in

a controllable way via the magnetic properties of the DMS system. This can

be done e.g., by choosing different host materials or by variation of the Mn2+

concentration or temperature. Due to the interaction of the free carriers with

the Mn2+ ions there is an additional term in the Zeeman splitting:

EZ = Eintr.Z + EGiant

Z = gµBB + xS0N0αe/hB5/2(5µBgMnB

2kB(TMn + T0)). (21)

The first term on the right-hand side corresponds to the intrinsic Zeeman split-

ting (see Eq. (12)). The second term stems from the exchange interaction of

electrons (holes) with the Mn2+ ions and is often referred to as Giant Zeeman

term. Herein, x is the effective average Mn2+ concentration, N0αe/h is the

exchange integral for electrons (holes) and B5/2(ξ) is the modified Brillouin

function [18,41]. This modified Brillouin function describes the magnetization

behavior of the manganese spin system. It is characterized by a strong temper-

ature dependency and a saturating behavior at high external magnetic fields

and low temperatures [18]. gMn is the effective Lande factor of manganese, kB

the Boltzmann constant and TMn the temperature of the Mn2+ spin system.

S0 and T0 take into account the Mn-Mn antiferromagnetic interaction. The

MPGE yields a photocurrent proportional to the Zeeman splitting (jZ ∝ EZ),

consequently all the characteristics of the studied photocurrents can be ex-

plained by Eq. (21) e.g., the drastic change of the amplitude with the temper-

ature, the change of sign (in case of Cd(Mn)Te) and the saturation. All these

findings result from the interplay of intrinsic and Giant Zeeman contribution,

depending on which part dominates.

For high temperatures Eintr.Z dominates. Hence, the current has a pure linear

dependency on By and only a weak dependency on T . By contrast, for low

temperatures the current is driven by EGiantZ which saturates for high mag-

netic fields. In the case of Cd(Mn)Te these two contributions have different

signs (g < 0, N0αe > 0) [18]. This results in the observed zero crossing (see

inset in Fig. 14 (a)). At this point both contributions have the same magni-

tude, thus Eintr.Z = EGiant

Z . For the In(Mn)As sample both terms are negative


Figure 15: (a) Magnetic field dependency of the calculated photocurrent

contributions Eintr.Z (dotted) and EGiant

Z (dashed, T = 1.5 K) for Cd(Mn)Te.

Here, Eintr.Z is assumed to be temperature independent. The sum of both

terms, EZ , is shown for temperatures of 1.5, 4.0, 20 and 150 K. For a better

comparison the curves of EZ for 150 K and Eintr.Z are multiplied by a factor of

5. (b) depicts the same calculated contributions for In(Mn)As. (c) presents

the temperature dependency of the calculated photocurrent contributions,

as well as the sum for a magnetic field value of 2 T. All curves are calculated

using Eq. (21) and the parameters from Tab. 3.

(g < 0, N0αh < 0) [59–62]. Hence, the photocurrent exhibits no change of sign

(see inset in Fig. 14 (b)).

Figure 15 depicts theoretical calculations of the photocurrent contributions for

both samples after Eq. (21). These are in very good agreement with the mea-

sured curves shown in Figs. 13 and 14. The curves are obtained for literature

values of parameters which are summarized in Tab. 3. Note, that the current

j is only proportional to EZ so that the definite sign of the measured photo-

current does not have to be the same as for the calculations (e.g., Figs. 13 (b)

and 15 (b)).


The observed increase of the photocurrent with a decrease of temperature can

not be explained by the Giant Zeeman effect alone: additional measurements

of photoluminescence for the Cd(Mn)Te sample yield a value for Zeeman spin

splitting of 2.6 meV for 3 T and 4.2 K. The intrinsic value, which is valid

for higher temperatures, is given by gµBB = -0.28 meV. Hence, the current

should increase by a factor of approximately 10 when the sample is cooled

down from e.g., 40 to 4.2 K at 3 T. By contrast, the THz radiation induced

photocurrent increases by a factor of approximately 100. This indicates that a

second mechanism plays a role for the current generation in DMS structures.

So far only the current generation based on the Zeeman splitting of the electron

subbands was considered (MPGE). However, there is a second mechanism for

the conversion of the electron fluxes j± which is specific for DMS structures:

the spin dependent scattering of carriers by polarized Mn2+ spins [18,63]. This

effect is described by an extra Hamiltonian which takes into account the elec-

tron and manganese ion interaction by a spin dependent scattering potential.

The two electron fluxes (j+ = −j−, from the zero bias spin separation) are

proportional to the momentum relaxation time τ±. By applying an exter-

nal magnetic field the manganese ions get polarized. This leads to different

scattering rates for spin-up and -down electrons and to unequal momentum

relaxation times τ+ 6= τ−. Consequently the fluxes do not compensate each

other anymore yielding an additional net electric current jSc. Assuming that

the electron scattering by the Mn2+ potential u is larger than the exchange

term described by α the second current contribution is given by [63]

jSc = 4eα

ujsSMn. (22)

Herein, SMn = −S0B5/2(ξ) is the average manganese spin along the magnetic

field direction.

Thus, the total net electric current in DMS structures is given by: j = jZ+jSc.

As both contributions have their origin in the polarization state of the Mn2+

spin system, the total current j will follow the Brillouin function.

It should be mentioned that for the In(Mn)As DMS sample a tiny positive

photocurrent was detected for T = 40 K (see Fig. 13 (b)). This observation

conflicts with the here presented theory. However, a possible explanation is the


interplay of the negative spin photocurrent of the MPGE and a positive orbital

photocurrent [64,65]. Further investigations are needed in order to verify this

explanation and is out of scope of this thesis.

4.3 Brief Summary

The results on terahertz radiation induced photocurrents in diluted magnetic

semiconductors clearly show that their microscopic origin is due to a spin de-

pendent asymmetric scattering of carriers, yielding a pure spin current. In the

presence of an external magnetic field this pure spin current can be converted

into a net electric current which is proportional to the Zeeman splitting of the

subbands. As the Zeeman splitting in DMS structures is drastically enhanced

due to the exchange interaction of free carriers with the manganese ions, the

currents receive a giant enhancement, too. In addition, the mechanism of cur-

rent generation is amplified by the spin dependent scattering of carriers on

polarized manganese ions.

5 Dirac Fermion System at Cyclotron Reso-

nance

In this chapter studies of cyclotron resonance assisted photocurrents induced

by terahertz radiation in HgTe quantum well systems will be presented. It

mainly refers to QW structures close to the critical width LC , where the en-

ergy spectrum is predicted to be characterized by a linear dependency. There-

fore, samples with a QW width of 6.6 nm were selected for the photocurrent

measurements. Complementary measurements on THz photoresistance and

radiation transmittance were also performed.

The focus of the measurements lies on cyclotron resonance and on the experi-

mental proof of the dispersion linear in k i.e., the results will be discussed in

detail by means of CR and band structure theory. Furthermore, a microscopic

model for current generation in a Dirac fermion system will be presented. The

experimental results of this chapter are published in Refs. [14, 15].

5.1 Terahertz Induced Photogalvanic Effects

In the following results for THz radiation induced photocurrents in HgTe QWs

without and with an external magnetic field will be presented. In both cases

the unbiased sample was illuminated under normal incidence with radiation

of a wavelength of λ = 118.8 µm at helium temperature. The experimental

geometry and signal detection is illustrated in Section 3.3.2.

Three different states of the 6.6 nm QW square shaped sample were studied:

the initial state with hole density p1 and two electron density states n1 and n2

achieved by optical doping [14]. For optical doping the sample in its initial state

is illuminated with a red light emitting diode (λ = 630 nm) for a finite time

t1 ≈ 15 s. By this illumination not only the carrier density but also the carrier

type changes to n1 due to the persistent photoconductivity effect [54,55]. Illu-

minating the sample further the carrier density saturates after approximately

80 s yielding carrier density n2. Typical results of magneto-transport measure-

ments can be seen in Fig. 8 (b) for the two different illumination states p1 and

n2. The figure shows well pronounced Shubnikov-de Haas oscillations in the

5 DIRAC FERMION SYSTEM AT CYCLOTRON RESONANCE 47

longitudinal resistance Rxx and Hall plateaus in the Hall resistance Rxy. The

change of carrier type can easily be seen by the different slopes of the Hall

resistance in the upper and lower panel.

5.1.1 Photocurrents at Zero Magnetic Field

The first measurements of photocurrents in HgTe QWs were performed for

samples with a width of 6.6 nm and linearly polarized THz radiation at zero

magnetic field. The polarization dependency on the direction of the electric

field plane was determined for all three carrier densities. Varying the angle α -

given by the linear polarization plane and the x direction, defined in the inset

of Fig. 16 (b) - an oscillating photosignal can be detected for x and y direction

which follows a sine/cosine dependency according to

Ui(α)/P = A+ Bsin(2α) + Ccos(2α), (23)

with i = x, y. This is shown for Uy/P in y direction in Fig. 16 (a). The fit

curves according to Eq. (23) are depicted in Fig. 16 as solid lines. The largest

signal is obtained for the carrier density n1 with an average photosignal of

0.91 mV/W. The photosignals of the sample states p1 and n2 have mean values

of 0.16 and 0.03 mV/W, respectively. For a better comparison some curves in

Fig. 16 (a) were scaled by factors indicated in the figure. The parameters of

the studied sample are presented in Tab. 4 giving information on e.g., carrier

densities and illumination time ti of optical doping. The fitting parameters A,

B, and C of Eq. (23) are given in Tab. 6.

In addition to the 6.6 nm QW square sample a structure with a QW width of

21 nm characterized by an inverted parabolic band structure was studied as a

reference for this part of the work. Similar to above the carrier density could

be increased by optical doping (carrier densities are given in Tab. 4) but for

this sample the conductance was n-type for both densities (n3 and n4). The

obtained results at zero magnetic field B as a function of the angle of the linear

polarization plane α is shown in Fig. 16 (b). Experimental conditions were the

same as for the 6.6 nm QW and again the photosignal shows an oscillating

behavior and can be fitted with Eq. (23). The mean values of the normalized

photosignal Uy/P are for both carrier densities in the same order of magnitude


Figure 16: THz radiation induced photosignal Uy, normalized by laser

power P , as a function of the angle α at 4.2 K and zero magnetic field for

the (a) 6.6 and (b) 21 nm QW square samples. Radiation with a wavelength

of 118.8 µm is applied normal to the QW layer. α describes the angle

between the linear polarization plane and the x direction, indicated in (b).

Solid lines are fit functions according to Eq. (23) with parameters given in

Tab. 6. Note, that some curves are scaled for a better comparison, indicated

in the figure.

at -0.016 (n3) and 0.011 mV/W (n4). Exact fitting parameters are summarized

in Tab. 6. Note, that the photosignal for n3 was multiplied by -1 for a better

comparison.

5.1.2 Cyclotron Resonance Assisted Photocurrents

Now photocurrents in the presence of an external magnetic field will be pre-

sented, namely under CR conditions. When illuminating the p-type 6.6 nm

QW square shaped sample with THz radiation and sweeping an external mag-

netic field B applied perpendicular to the QW plane a resonant photosignal

was detected. The photosignals Uy/P in y direction are shown for all types of

polarization and λ = 118.8 µm in Fig. 17. For linearly polarized radiation the


positions at which the well pronounced peaks occur are -0.44 and +0.40 T (see

black curve). The magnitudes for both magnetic field polarities are similar:

-9.93 and +7.98 mV/W for negative and positive B. When illuminating the

sample with right-handed circularly polarized light (σ+), the peaks stay at the

same magnetic field values, but the photosignal amplitude changes: while the

signal is increased for negative magnetic fields, it is decreased for positive B

values (see red curve in Fig. 17). In fact the resonant signal is about a factor

of 4 larger for negative than for positive magnetic fields. The situation for

left-handed circularly polarized light (σ−, blue line) is vice versa. The mea-

surements show that for a fixed radiation helicity (σ− or σ+) only one dominant

resonant peak appears - either for positive or negative B values. Moreover,

the normalized photosignal Uy/P which is shown in Fig. 17 is an odd function

Figure 17: Magnetic field dependency of the normalized photosignal Uy/P

for the p-type 6.6 nm QW square sample at T = 4.2 K. THz radiation

of λ = 118.8 µm was applied perpendicular to the sample with different

polarizations: for linearly polarized light (black curve) the peaks for negative

and positive B have approximately the same amplitude. By contrast, σ+

radiation (red curve) shows a much larger resonance for negative and σ−

radiation (blue curve) for positive magnetic field values.


of the magnetic field B. A similar behavior is observed for the photosignals

Ux/P in x direction (not shown). For reasons of simplification and because

the THz radiation induced photocurrent has no predefined direction due to

the low symmetry of the sample (C1 point group) [24], in the following only

the absolute value of the photosignal is presented. This is done according to

U/P =√

U2x + U2

y /P in which both measured voltage drops are considered, as

well as the normalization by the laser power P .

Figure 18: Magnetic field dependency of the normalized photosignal U/P

for the 6.6 nm QW square sample with three different carrier densities.

Right-handed circularly polarized radiation with λ = 118.8 µm was applied

normal to the QW layer at T = 4.2 K.

This normalized, absolute photosignal is shown for the p-type sample and for

σ+ radiation in Fig. 18 (red curve). Only the data in the vicinity of the

pronounced peak position Bc is shown, here at a negative magnetic field value

of -0.44 T. The maximum signal Umax = 18.47 mV/W is by two orders of

magnitude larger than at zero magnetic field UB=0 = 0.15 mV/W. When the

carrier density is increased by optical doping to a value of n1 ≈ 2 × p1, the

situation changes dramatically. For right-handed circularly polarized light the


resonant photosignal position Bc jumps from negative to positive magnetic

fields to a value of +0.70 T (see Fig. 18, black curve). The amplitude of U/P

for n1 has a similar maximum amplitude of 16.76 mV/W, as for p1. Repeating

the same measurement for a carrier density of n2 ≈ 3 × n1 the peak appears

at +1.20 T with a peak magnitude of 8.14 mV/W (blue curve). For left-

handed circularly polarized light the situation is vice versa: while the p-type

sample shows a resonant photosignal at positive magnetic fields (+0.38 T),

both n-type samples show peaks at negative B values (-0.78 and -1.20 T).

These resonance positions are given in Tab. 4 as a mean value of |Bc| for bothradiation helicities.

In order to characterize the origin of the resonance phenomenon, additional

measurements of radiation transmittance and photoresistance (see Section 5.2)

were carried out. The setup is illustrated in Section 3.3.4. The results for cir-

cularly polarized radiation at a wavelength of λ = 118.8 µm and T = 4.2 K

are depicted in Fig. 19. Sweeping the magnetic field a distinct peak in the

normalized transmitted radiation was detected for all three carrier densities.

The peak positions correspond to the magnetic field values at which the condi-

tion for CR is fulfilled. The most pronounced peak is observed for n2. In that

case the normalized transmittance decreases by approximately 25%, whereas

LW (nm) carrier density (cm−2) ti (s) BPGEc (T) BTM

c (T)

p1 6.6 1.5 × 1010 (p-type ) - 0.41 0.42

n1 6.6 3.4 × 1010 (n-type) 15 0.74 0.74

n2 6.6 11 × 1010 (n-type) 80 1.20 1.16

n3 21 18 × 1010 (n-type) - 2.28/2.74 2.31/2.91

n4 21 24 × 1010 (n-type) 80 2.38/2.74 2.31/2.88

Table 4: Carrier densities and types of the 6.6 and 21 nm QW square

samples. The parameters were obtained at 4.2 K for different illumination

states of the sample. ti is the duration of the illumination by a red light

emitting diode, used for the optical doping. The last two columns present

the magnetic field values of the resonance position Bc, observed in the pho-

tocurrent and transmission measurements. Magnetic field positions have

been averaged over the absolute value of Bc for σ+ and σ− measurements.


Figure 19: Normalized transmittance of circularly polarized radiation as

a function of the magnetic field B for the 6.6 nm QW square sample with

three different carrier densities. Radiation with λ = 118.8 µm was applied

normal to the QW layer at T = 4.2 K. Note, that the curves of p1 and n1

scale on the left, while the signal for n2 scales on the right. Solid lines are

Lorentzian fits according to Eq. (24) with parameters given in Tab. 7. In

all three cases a resonant peak can clearly be seen, corresponding to the

magnetic field value, where the cyclotron resonance condition is fulfilled.

for p1 and n1 the signal only drops by 4-5%. For this reason the radiation

transmission curves for p1 and n1 scale on the left and the one for n2 scales

on the right. The center positions BTMc of the observed resonances are -0.42,

+0.72 and +1.17 T for σ+ radiation and carrier densities of p1, n1 and n2,

respectively. The situation for σ− is vice versa, yielding resonance positions

of +0.42, -0.75 and -1.14 T. The center positions are summarized in Tab. 4 as

an average of the absolute values of positive and negative cyclotron resonance

position. All radiation transmittance data were fitted by Lorentzian curves

given by

F(B) = o± 2a

π

w

w2 + 4(B − Bc)2, (24)

with offset o, amplitude a, width w and center position Bc, shown as solid lines

in Fig. 19. Exact fitting parameters of Eq. (24) are given in Tab. 7.


A 21 nm QW square sample, characterized by an inverted parabolic band struc-

ture, was investigated by THz radiation induced photocurrents and radiation

transmittance. The photosignal is shown for σ+ radiation with a wavelength of

λ = 118.8 µm as a function of the magnetic field B in Fig. 20 (a). For the ini-

tial carrier density n3 = 18×1010 cm−2 two peaks are observed at much higher

magnetic fields Bc than for the 6.6 nm QW sample: +2.30 and +2.78 T. For

negative magnetic fields the resonant signal is less pronounced and by about

a factor of 3 smaller. Therefore, the curves are limited to positive magnetic

fields. Increasing the carrier density by optical doping to n4 = 24× 1010 cm−2

almost no shift of resonance position is observed (Bc = +2.44 and +2.78 T),

in contrast to the behavior of the 6.6 nm sample. For σ− polarized light the

peaks occur at -2.26 and -2.70 T for n3 and at -2.32 and -2.70 T for n4.

Figure 20: (a) Normalized photosignal U/P as a function of the normal

applied magnetic field B for the 21 nm QW square sample with two different

carrier densities. The sample was illuminated by right-handed circularly

polarized light of normal incidence with λ = 118.8 µm at T = 4.2 K. (b)

shows the corresponding measurement of radiation transmittance. Solid

lines in (b) are Lorentzian fits according to Eq. (24) with parameters given

in Tab. 8.


The results for radiation transmittance are similar to those of the photocur-

rent measurements and are illustrated in Fig. 20 (b) for right-handed circularly

polarized light: the splitting of the resonance is clearly resolved at 2.31 and

2.91 T for n3 and at 2.31 and 2.88 T for n4. The normalized transmittance

decreases by ≈ 15% for the first and by ≈ 25% for the second resonant peak.

They can be nicely fitted by two Lorentzian curves using Eq. (24). Exact

fitting parameters are given in Tab. 8. For negative magnetic fields no cyclo-

tron resonance is detected; for left-handed circularly polarized radiation the

situation is vice versa.

The resonance positions estimated from photocurrent and transmittance mea-

surements are given in Tab. 4 as an average value of |Bc| for both radiation

helicities.

5.2 Photoresistance Measurements

The effect of cyclotron resonance has also been studied by THz photoresistance,

which supports the measurements of radiation transmittance of the last sec-

tion. These measurements were performed in close cooperation with Dimitry

Kvon1 and Vasily V. Bel’kov2. Two Hall bar structures with different HgTe

QW widths LW = 6.4 and 6.6 nm were investigated; both are close to the

predicted critical width LC [11]. The setup was according to the description

in Section 3.3.3 and the obtained results are published in [14].

1Institute of Semiconductor Physics, Novosibirsk, Russia2Ioffe Institute, St.Petersburg, Russia

LW (nm) carrier density (cm−2) Bc (T)

n5 6.6 1.7 × 1010 0.29

n6 6.6 7.2 × 1010 1.03

n7 6.4 1.8 × 1010 0.40

Table 5: Carrier densities of the 6.4 and 6.6 nm QW Hall bar samples

obtained at 4.2 K by magneto-transport measurements and corresponding

magnetic field values Bc of the cyclotron resonance position.


Illuminating the biased 6.6 nm QW Hall bar sample in its initial state n5 with

THz radiation and sweeping an external magnetic field a resonant change of the

resistance was observed. This is shown in Fig. 21 (a) for linearly polarized THz

radiation of λ = 118.8 µm. The resonance with an amplitude of ∆Rx = 2.73 Ω

is detected at a magnetic field value of +0.29 T and corresponds to the position

of cyclotron resonance. When the initial carrier density is increased to a value

of n6 ≈ 5× n5 by optical doping the cyclotron resonance position Bc shifts to

a value of +1.03 T. Here ∆Rx is given by 4.88 Ω.

The 6.4 nm Hall bar sample with carrier density n7 shows a change of resistance

of ∆Rx = 2.36 Ω at a CR position of +0.40 T. This is shown in Fig. 21 (b).

The carrier densities and the corresponding CR positionsBc are given in Tab. 5.

All curves can be properly fitted by a Lorentzian curve according to Eq. (24),

Figure 21: Magnetic field dependency of THz photoresistance ∆Rx at

4.2 K and for λ = 118.8 µm. (a) shows the results for the 6.6 nm QW Hall

bar sample with two different carrier densities n5 and n6. (b) presents ∆Rx

for the 6.4 nm QW Hall bar sample with carrier density n7. Solid lines are

Lorentzian fits according to Eq. (24).


depicted in Fig. 21 as solid lines. The exact fitting parameters are given in

Tab. 9.

5.3 Discussion

The resonant THz radiation induced photocurrents of Section 5.1.2, as well as

the surprising dependency of the resonance position on the carrier density are

the main topic of this discussion. The same dependency was observed in radi-

ation transmittance (see Section 5.1.2) and photoresistance (see Section 5.2)

measurements, too. These results will be analyzed by cyclotron resonance the-

ory and with reference to the unique band structure of HgTe (see Section 2.4

and 2.1). Furthermore, a microscopic model of current generation in systems

characterized by a linear band structure will be presented which was devel-

oped parallel to the experimental work by Grigory V. Budkin3 and Sergey A.

Tarasenko3 [15]. But before the CR assisted photocurrents will be discussed,

photocurrents at zero magnetic field will be briefly addressed.

In the following only the main theoretical outcome of Ref. [15] will be intro-

duced, as a detailed derivation of the equations is out of scope of this thesis.

5.3.1 Photocurrents at Zero Magnetic Field

Illumination of a 6.6 and 21 nm HgTe QW square sample with linearly polar-

ized THz radiation results in a measurable electric current. When the angle of

the linear polarization plane is varied, the photosignal oscillates as a function

of sine and cosine (see Fig. 16). All curves can be well fitted by Eq. (23).

Exact parameters of A, B and C (offset and prefactors of sin2α and cos2α)

are given in Tab. 6. While such a current has not yet been detected in 6.6 nm

HgTe QWs, it was observed earlier for 21 nm HgTe QWs (see Refs. [24, 66]).

Herein, the current is shown to be caused by the circular and linear photogal-

vanic effect (for reviews see Refs. [17, 67]). Considering a parabolic dispersion

and a C1 point symmetry, the dependency of the photocurrent on the linear

polarization state for normal incidence of radiation and zero magnetic field is

3Ioffe Institute, St.Petersburg, Russia


given by [24]

jx =

[

Xxxysin(2α)−Xxxx +Xxyy

2+

Xxxx −Xxyy

2cos(2α)

]

Iη,

jy =

[

Xyxysin(2α)−Xyxx +Xyyy

2+

Xyxx −Xyyy

2cos(2α)

]

Iη,(25)

with third rank tensor X, angle of linear polarization α, radiation intensity

I and radiation absorption coefficient η. All components of X are linearly

independent and may be nonzero, due to the low symmetry of the (013) grown

HgTe QWs. Just as in Ref. [24] these equations correspond exactly to the po-

larization dependency of the photosignal of the 21 nm QW sample (see Fig. 16

(b)). Moreover, they also describe the dependency for the 6.6 nm QW sample

very well (see Fig. 16 (a)). For this reason a similar mechanism for the cur-

rent generation can be expected in both samples, but this would need further

measurements and analysis and is out of scope of this thesis.

LW (nm) A B C

p1 6.6 0.16 0.06 0.04

n1 6.6 0.91 -0.09 -0.14

n2 6.6 0.03 -0.01 -0.02

n3 21 -0.016 0.001 -0.010

n4 21 0.011 0.002 -0.003

Table 6: Fit parameters of Eq. (23) for the photosignal as a function of α

for the 6.6 and 21 nm HgTe QW sample (see Fig. 16). Note, that scaling

factors are not included in this table.

5.3.2 Photocurrents and Photoresistance under Cyclotron Reso-

nance Condition

For a magnetic field applied normal to the quantum well plane the THz ra-

diation induced photosignals show a resonant behavior. This can be seen in

Fig. 18 for the 6.6 nm QW sample for all investigated carrier densities p1, n1

and n2. The observed peaks coincide very well with the positions of cyclo-


tron resonance in corresponding measurements of radiation transmittance (see

Fig. 19). For an easy comparison both resonance positions BPGEc taken from

photocurrent and BTMc obtained by transmission measurements, are summa-

rized in Tab. 4. The average of the absolute value for positive and negative

magnetic fields is listed. This good agreement indicates that the photocurrent

is cyclotron resonance assisted. Additional support for this assumption comes

from the dependency of the photosignal amplitude on the light’s helicity for a

certain carrier type. In Fig. 17 (hole density p1) the peaks for the σ+ photosig-

nal are much more pronounced for negative than for positive magnetic fields.

For radiation transmittance measurements the peaks arise for one helicity and

magnetic field polarity only (not shown). This is in accordance with the theory

of cyclotron resonance (see Section 2.4 and Eq. (16)): when radiation helicity

and carrier type of a two dimensional electron (hole) gas are fixed, the power

absorption of THz radiation by free carriers is only allowed for one magnetic

field polarity.

The measurement of photoresistance is - next to radiation transmission - an

established method for investigating the effect of cyclotron resonance. There-

fore, it can be assumed that the peaks in the photoresistance (see Fig. 21)

correspond to the magnetic field positions of CR.

The peak positions in Fig. 21 (a) for the 6.6 nm QW Hall bar sample show

the same extraordinary dependency on the carrier density n, as the CR as-

sisted THz radiation induced photocurrents and radiation transmittance (see

Figs. 18 and 19). In Fig. 22 (a) the center positions Bc of the photocurrent

and photoresistance peaks are plotted as a function of the carrier density n,

as square and dot data points.

Besides this surprising dependency of Bc on n, the magnetic field values of

CR are unusual small. Taking into account the used photon energy E~ω =

10.35 meV and the effective mass of electrons in HgTe QWs ofm∗/me ≈ 0.026 -

0.034 [68], the magnetic field position of CR can be calculated according to [6]

Bc =E~ωm

∗

~e, (26)

yielding values for Bc between 2.34 and 3.02 T. By contrast, the experimental

values are significantly smaller, they are all in the range of |Bc| = 0.41 - 1.2 T.


A possible explanation for these unexpected observations (shift of Bc with

carrier density and small effective masses) is the special energy spectrum of

HgTe QW samples with the critical width LC . This type of structure should

have according to Refs. [10, 13] a linear energy spectrum with massless two

dimensional Dirac fermions. Indeed, in such systems the cyclotron frequency

is given by the well known expression ωc = eB/mc. Herein, mc is the effective

cyclotron mass at Fermi energy EF . It is given by [69,70]

mc =EF

v2DF

, (27)

∝

Figure 22: (a) Dependency of the resonance position Bc on the carrier

density of the 6.6 nm QW samples. Values are taken from photoresistance

(Hall bar sample, dots) and photocurrent (square sample, squares) mea-

surements. The data show a square root dependency according to Bc ∝√n

(black line). (b) Splitting of Landau levels (LL) as a function of the mag-

netic field B. Zeeman splitting is neglected. Black arrows indicate a fixed

photon energy E~ω. In order to fit this energy to two different transitions

(e.g., from 2nd to 3rd and from 3rd to 4th Landau level for Fermi energy EF1

and EF2, respectively), the magnetic field has to be increased.


with the constant velocity vDF of Dirac fermions. Taking into account the

Fermi energy of massless two dimensional Dirac fermions [15]

EF =√2πn(~vDF ) (28)

the CR position Bc for a linear dispersion can be expressed as

Bc = ~ω

√2πn

evDF

. (29)

This dependency can be nicely seen in Fig. 22 (a). Even though the data

points were obtained in different types of measurements on different samples

and designs all data can be well described with a single curve proportional to√n (Eq. (29)). The origin of this square root dependency can be found in the

Landau level splitting of systems with a Dirac-like dispersion. In this special

case the l-th Landau level is not given by the standard expression of Eq. (18)

but by [13,71–73]

El = sgn(l)vDF

√

2 |l| ~eB, (30)

where l ∈ N. Accordingly the Landau level splitting is not equidistant and is

estimated to be

∆El,l+1 = vDF

√2~eB(

√l −√

(l + 1)), (31)

for the l-th and (l+1)-st Landau level.

The square root dependency of the l-th Landau level El on the magnetic field

B (Eq. (30)) is shown schematically in Fig. 22 (b). For a carrier density n1 with

corresponding Fermi energy EF1 a transition from second to third Landau level

is induced by radiation with a fixed photon energy E~ω. The photon energy

is indicated by the black arrow. The magnetic field at which this transition

occurs is Bc1 according to Eq. (31), with ∆El,l+1 = E~ω. When the carrier

density is increased to n2 (Fermi energy EF2) a transition from second to third

Landau level is not possible anymore as the latter is completely populated for

Bc1. Hence, the position of CR has to shift to a higher magnetic field value Bc2

in order to fulfill Eq. (31). Consequently the observed shift of the CR position

Bc with variation of the carrier density n is a direct evidence that the energy


spectrum of the studied samples with LW = 6.6 nm is that of two dimensional

Dirac fermions.

On the basis of these observations the constant electron velocity vDF can be

calculated according to Eq. (29). The obtained value of 7.17×105 m/s is almost

constant for the different densities. The value is in good agreement with the

electron velocity calculated in Ref. [13]. Moreover, the cyclotron masses mc

of the 6.6 nm QW samples can be calculated according to Eq. (27). They

are extraordinary small with values in the range of 0.003 to 0.012 me. They

also show - similar to Bc - a square root dependency on the carrier density

according to Eq. (27) (not shown).

The significant higher resonance positions of approximately 2.30 and 2.90 T

measured for the 21 nm QW sample (Fig. 20) are in accordance with the

effect of CR for a QW system with parabolic dispersion. These CR positions

correspond to effective masses of 0.026 and 0.032 me obtained by Eq. (26) and

lie within the range given in Ref. [68]. The negligible dependency of Bc on the

carrier density fit to the CR behavior of a normal parabolic dispersion with

equidistant Landau level splitting, given by ∆El,l+1 = ~ωc. In this case the

position of the resonance is independent of the Fermi energy as all transitions

have the same energetic splitting (compare Section 2.4).

The observed two cyclotron resonance peaks measured by photocurrents and

radiation transmittance for this 21 nm QW sample can be due to different

effects (see Fig. 20). A similar splitting has also been observed in Ref. [74]

in radiation transmittance experiments on a 15 nm HgTe QW. A possible

explanation could be a second occupied subband and nonparabolic effects of

the band structure [75]. Additional analysis is needed to isolate the origin of

this splitting and is out of scope of this thesis.

It should be noted that a decrease of the QW width from 6.6 to 6.4 nm results

in a noticeable shift of the resonance position to higher magnetic fields (from

+0.29 to +0.40 T, see Fig. 21). This behavior allows the assumption that

the latter QW has a narrow energy band gap with Dirac fermions of nonzero

mass [14]. The critical width at which a Dirac-like energy spectrum should

occur for HgTe QWs is according to Ref. [11] 6.4 nm. In this work the sample

with LW = 6.6 nm clearly shows properties of a Dirac-like linear band structure.


This small discrepancy in the critical width - 6.6 nm vs. 6.4 nm - may result

from e.g., the different growth directions (the investigated samples in this work

are (013) grown, whereas the sample in Ref. [11] is (001) grown), possible

strain [76], layer design or substrate (here: GaAs, Ref. [11]: CdTe).

5.3.3 Microscopic Picture of Current Generation

On a first view the enhancement of the photosignals in Figs. 17 and 18 seems

to stem from cyclotron resonance. For linearly polarized radiation the res-

onant photosignal shows an odd dependency on the external magnetic field

(see Fig. 17). However, CR is an even function in B. This can be seen with

Eq. (16) by the comparison of radiation absorption at CR position with the

absorption far outside the resonance e.g., ω = 0. From that comparison follows

that radiation absorption is given by a term ∝ (ωcτ)2 ∝ B2.

The momentum scattering time τ can be estimated from the full width w at

half maximum of the transmission measurements according to τ = 2m∗/we,

which can be derived with the help of Eqs. (16) and (24). For the 6.6 nm QW

sample with carrier density p1 (see Fig. 19) this gives values of τ ≈ 0.15 ps

and (ωcτ)2 = 5.69. This means that at CR position the photosignal should be

enhanced by a factor of 5.69 compared to zero magnetic field.

By contrast, the measured signal at resonance positionBc = −0.44 T is approx-

imately two orders of magnitude larger than the signal at zero magnetic field

(Umax/P = 18.47 mV/W and UB=0/P = 0.16 mV/W, see Figs. 16 and 18).

This large discrepancy leads to the conclusion that the resonant photosignal

is not the signal at zero magnetic field enhanced by cyclotron resonance but

implies that it is due to another microscopic root cause.

It is now shown that all observed experimental features can be well understood

in the framework of the magnetogyrotropic photogalvanic effect (compare Sec-

tion 2.2). This effect is well known from other systems like diluted magnetic

semiconductor structures. Detailed results for DMS samples were reported

in Chapter 4. Herein, a current was observed when the sample was illumi-

nated with THz radiation and an external magnetic field was applied. The

origin of this current is a spin dependent asymmetric scattering of carriers


in the Zeeman spin split parabolic subbands. The main difference to HgTe

QW structures with a width of 6.6 nm is the fact that the latter has a linear

band structure. However, the model of the MPGE can be adopted to a linear

dispersion as well, which will be illustrated in the following.

Figure 23: (a) and (b) show the spin-down and -up Dirac cone. The

Fermi energy EF indicates up to which energy the cones are populated

in equilibrium. The relaxation of the carriers heated by THz radiation is

spin- and k-dependent, yielding an electron flux j± in each spin cone. The

different relaxation rates for positive and negative wave vectors are indicated

by the arrows’ thickness. c) By applying an external magnetic field the cones

shift apart energetically. This Zeeman splitting is indicated by ∆EZ and

results in an imbalance of the population of the two cones. Consequently,

j− and j+ have different strengths.

The absorption of THz radiation results in a strong heating of the two di-

mensional electron (hole) gas. The matrix element of the electron-phonon

interaction has an additional term for gyrotropic media (e.g., HgTe QWs [31])

which is proportional to the spin and wavevector i.e., ∝ σα(kβ + k′

β), with

Pauli matrices σ and initial (scattered) wavevector k (k′) (see Eq. (10)). Con-

sequently, the energy relaxation of the carriers heated by THz radiation in each

spin cone is asymmetric for positive and negative k (see Fig. 23 (a) and (b) for

the spin-down and -up cone). Considering e.g., the spin-down cone (σ < 0),

Eq. (10) yields a higher probability Wk,k′ ∝ |Vk,k′ |2 for scattering events of elec-trons with phonons for negative than for positive k. Thus, more spin-down

electrons with a negative wavevector move back towards k = 0 resulting in


an electron flux j− in negative k direction. The situation for the spin-up cone

(σ > 0) is vice versa with a flux j+ in positive k direction. The two electron

fluxes are oppositely directed and of equal strength (j+ = -j−), hence, they

cancel each other out. The total net electric current is zero but a pure spin

current is formed defined by Eq. (11).

This pure spin current can be converted into a net electric current by the

application of an external magnetic field: the two spin cones split up in energy

due to the Zeeman effect, which is illustrated in Fig. 23 (c). One spin cone

is now preferentially populated. This results in an imbalance of the electron

fluxes and gives rise to a measurable net electric current jZ = j− + j+.

Assuming that EF ≫ kBT , EF ≫ ∆EZ/2 and ωcτ ≫ 1 the magnitude of the

current can be estimated to [15]

|jZ | ∝evDF

EFωc

∆EZ

EF

ξIη, (32)

where ξ takes into account the spin orbit interaction strength and η the ab-

sorption in the vicinity of CR given by

η ∝ EF τ

~

1

1 + (ω − ωc)2τ 2. (33)

Equations (32) and (33) are valid for both parabolic and linear dispersions.

The only difference exists in prefactors which are omitted here. Equation (33)

contains the typical Lorentzian behavior of the cyclotron resonance, which can

be seen in all measured signals: the photosignals clearly follow the spectral

behavior of η (see e.g., Fig. 18). Furthermore, the current jZ given by Eq. (32)

is proportional to the Zeeman splitting ∆EZ = gµBB and is therefore an odd

function in the magnetic field, as shown in Fig. 17. Moreover, the current also

depends on the spin orbit interaction strength ξ. Since the investigated samples

are based on HgTe, which is characterized by a strong spin orbit coupling and

a large effective g factor [77], the current is additionally enhanced by these

properties. This explains the increase by two orders of magnitude, observed

in the experiment on cyclotron resonance assisted photocurrents in Fig. 18 for

the p-type sample, whereas the radiation absorption only gives a factor of ≈6.


For the experimental setup, where the signals are measured via the voltage drop

over a 10 MΩ load resistor which is much higher than the sample resistance,

the photosignal is given by [15]

U =∣

∣

∣jZ

ωcτa

σ

∣

∣

∣, (34)

with the sample size a and zero field conductivity σ = e2EF τ/(2π~2). For a

fixed radiation wavelength the dependency of the photosignal amplitude on

the carrier density is estimated to [15]

U ∝ gτ√nv2DF

. (35)

This is in accordance with the results shown in Fig. 18: by increasing the ab-

solute value of the carrier density by approximately a factor of 7 (from p1 to

n2) the maximum amplitude of the THz radiation induced photosignal U/P

decreases only by a factor of 2.3. This value is in good agreement with the

factor√7 = 2.7 which is obtained with Eq. (35). Thus, the above introduced

microscopic model and theory can explain all experimental results of this chap-

ter.

5.4 Brief Summary

The linear, Dirac-like character of the band structure of 6.6 nm HgTe QW

structures could be verified by means of THz radiation induced photocurrents,

THz radiation transmittance, and THz photoresistance assisted by cyclotron

resonance. The observed shift of the CR position with variation of the carrier

density (see Figs. 18, 19 and 21) can be explained unambiguously with the

nonequidistant Landau level splitting of a Dirac fermion system, characterized

by a linear dispersion.

In addition a microscopic model based on spin dependent scattering of electrons

was introduced for the current generation in Dirac cones, which is related

to the magnetogyrotropic photogalvanic effect described in Chapter 4. This

microscopic model and the corresponding equations can explain all observed

features - especially the large signals - of the photocurrent measurements.


The photocurrent described by Eq. (32) is proportional to the spin orbit in-

teraction strength, the radiation absorption and the Zeeman splitting. All

three contributing factors are enhanced in the system under study due to the

measurement geometry and the material properties of the samples. Conse-

quently, the measured signals are extraordinary large compared to those of

conventional semiconductor structures. Considering the sample’s resistance

according to I = U/R e.g., for the n2 carrier density of the 6.6 nm HgTe QW

sample, a current of 7.3 µA/W was measured for a magnetic field of +1.2 T.

By contrast, for the Cd(Mn)Te DMS sample a maximum current was observed

of only 0.69 µA/W for 3 T (see Fig. 13). Also for GaAs QW structures the

typical magnitude of photocurrents is much less and lies in the range of 10 -

100 nA/W [78].

Furthermore, according to Eq. (32) these currents are proportional to the mag-

netic field, a dependency that is actually observed (see Fig. 17). The Lorentzian

like behavior of these photocurrents stems from the spectral behavior of the

absorption η given in Eq. (33). Furthermore, the rather weak decrease of the

photosignals’ maximum amplitude with the increase of the carrier density from

p1 and n2 (see Fig. 18) is in accordance with Eq. (35).


offset o center xc width w amplitude a

p1 1.02 0.40 0.34 0.03

n1 1.00 0.70 0.34 0.03

n2 1.02 1.18 0.34 0.13

Table 7: Parameters of the Lorentzian fit curves according to Eq. (24) for

the radiation transmission measurements of the 6.6 nm QW square sample

(solid lines in Fig. 19).

peak 1 offset o center1 xc width1 w amplitude1 a

n3 0.95 2.31 0.20 0.04

n4 0.98 2.33 0.20 0.05

peak 2 center2 xc width2 w amplitude2

n3 2.93 0.26 0.09

n4 2.89 0.26 0.12

Table 8: Parameters of the Lorentzian fit curves according to Eq. (24) for

the radiation transmission measurements of the 21 nm QW square sample

(solid lines in Fig. 20 (b)).

LW (nm) offset o center xc width w amplitude a

n5 6.6 -0.13 0.30 0.09 0.41

n6 6.6 -0.52 1.04 0.21 1.75

n7 6.4 0.11 0.41 0.17 0.64

Table 9: Parameters of the Lorentzian fit curves according to Eq. (24)

for the photoresistance measurements of the 6.6 and 6.4 nm QW Hall bar

samples (solid lines in Fig. 21).

6 Quantum Oscillations in Mercury Telluride

Quantum Wells

In this chapter the studies of quantum oscillation phenomena in HgTe based

QW structures are presented and analyzed. The oscillations are detected in

photocurrents and in the change of resistance, when illuminating the sample

with terahertz radiation and sweeping an external magnetic field. As an im-

portant feature their envelope shows a nonmonotonic behavior. In order to get

a good general view of these observations different kind of band structures were

studied. This was achieved by investigating HgTe QWs with different widths

ranging from 5 to 21 nm. Hence, all types of dispersion were available: nor-

mal and inverted parabolic, as well as a linear band structure. To explore the

origin of the oscillations and the nonmonotonic behavior of their envelope func-

tion, the measurements of photocurrents are accompanied by measurements of

photoresistance, magneto-transport and radiation transmission. The data are

discussed on the basis of a microscopic model for current generation which

gives a different view on the magnetogyrotropic photogalvanic effect, already

introduced in the previous chapters. Quantum transport phenomena like the

Shubnikov-de Haas and de Haas-van Alphen effect will also be considered in

order to explain the mechanism of the oscillations. The results of this chapter

are published in Ref. [20]. Note, that all photocurrent and photoresistance

data i.e., their oscillations, are fully reproducible.

6.1 Experimental Results

All investigated samples of this chapter were characterized by standard magneto-

transport measurements. The corresponding carrier densities, band stucture

and design are summarized in Tab. 10. Furthermore, the samples are consec-

utively numbered. In particular, some samples (#3, #7) were already investi-

gated in Chapter 5. First the data obtained for the 8 nm HgCdTe QW Hall

bar sample #2 will be presented, which show the most pronounced oscillations

in the photosignal and photoresistance. After that it will be shown that the os-

6 QUANTUM OSCILLATIONS IN MERCURY TELLURIDE QWS 69

cillations can be found in square shaped, pure HgTe QW structures of various

widths, too.

sample LW (nm) band structure carrier density (cm−2) design ti (s)

#1 5 normal 24 × 1010 square -

#2∗ 8 normal 75 × 1010 Hall bar -

#3 6.6 linear 11 × 1010 square 80

#4 6.6 linear 0.5 - 45 × 1010 Hall bar -

#5 8 inverted 24 × 1010 square -

#6 8 inverted 31 × 1010 square 80

#7 21 inverted 17 × 1010 square -

Table 10: QW width LW , band structure, carrier density (obtained at

4.2 K) and design of the investigated samples. ti is the time of illumination

with a red LED used for optical doping. Note, that samples #5 and #6

are the same but in two different illumination states. ∗The QW contains

cadmium: Hg0.86Cd0.14Te.

6.1.1 Results for a Parabolic Dispersion

Exciting the 8 nm HgCdTe QW Hall bar sample #2 with THz radiation

and sweeping an external magnetic field a signal was detected which shows

a nonmonotonic behavior superimposed by oscillations. The magnetic field

and the propagation direction of the linearly polarized THz radiation with

λ = 118.8 µm were aligned normal to the QW plane. This photosignal Ux/P

is depicted in Fig. 24 (a) for T = 4.2 K. It features a maximum value at

+2.78 T (-3.16 T for negative magnetic fields, not shown). Figure 24 also shows

the corresponding longitudinal resistance with well pronounced Shubnikov-de

Haas oscillations in black, scaled on the right. It can easily be seen that the

oscillations in the photosignal follow the SdH oscillations quite nicely: when

numbering the minima positions of the photosignal consecutively from 1 (min-

imum at ≈ 6.17 T) to 7 (minimum at ≈ 2.5 T) and plotting these as a function

of the inverse magnetic field 1/B they can be nicely fitted by a straight line

with a slope of 26.2 T. If the filling factors υ (see Section 2.5) of the SdH

oscillations are displayed in the same manner they also show a linear behavior


on the inverse magnetic field with a very similar slope of 30.6 T. The good

correlation between the two measurements is shown in Fig. 24 (c), where both

filling factors υ are plotted vs. the inverse magnetic field. Note, that the signal

Uy/P along the y direction shows a similar result. As this is the case for all

measurements described in this chapter the presented figures are limited to

one direction of voltage drop only (casually the x direction).

Figure 24: Results obtained for the 8 nm HgCdTe QW Hall bar sample

#2 excited by linearly polarized radiation of λ = 118.8 µm at T = 4.2 K.

(a) shows the photosignal Ux, normalized by radiation power P induced in

the unbiased sample. (b) presents the photoresistance ∆Rx for the biased

sample. Black dashed curves show SdH oscillations of the longitudinal resis-

tance Rxx which scale on the right; gray dotted lines show the calculated CR

absorption profile according to Eq. (45) for two different scattering times

τ (1: 0.74 ps, 3: 0.3 ps, see Section 6.2.2). Filling factor υ vs. inverse

magnetic field 1/B is depicted in (c) for SdH and photosignal minima. The

latter is numbered consecutively with integers from 1 to 7.


A similar dependency is observed in the same sample for the THz photore-

sistance ∆Rx shown in Fig. 24 (b). A maximum signal ∆Rx is observed for

-3.14 and +3.12 T (again only the positive magnetic field range is shown).

While oscillations in the photoconductivity with similar features have already

been detected in HgTe based and other QW systems [79–81], oscillations in

the photocurrent have not been observed so far.

To summarize, for both measurements, photocurrent and photoresistance, os-

cillations and a nonmonotonic behavior of their envelope are observed i.e., the

envelope has a distinct maximum. The results of Chapter 5 suggest that this

maximum might stem from the effect of cyclotron resonance. In order to ver-

ify this assumption, additional measurements were performed on large area

square shaped samples. The measurements of photocurrent and photoresis-

tance for various widths LW of pure HgTe QWs with a normal and inverted

parabolic dispersions all show a similar behavior like sample #2. The results

are shown in Figs. 25 - 27 with corresponding radiation transmittance and

magneto-transport curves.

Figures 25 (a) and 26 (a) depict the results of the photocurrent and photoresis-

tance measurement for the 8 nm pure HgTe QW sample #5. Both curves were

obtained for σ+ radiation with λ = 118.8 µm at liquid helium temperature.

The photosignal shows pronounced oscillations and is enhanced in the vicinity

of cyclotron resonance. The position of CR at about Bc = 1.8 T is detected

by radiation transmission which is shown in Fig. 25 (b). The photosignal ex-

hibits a maximum signal of 1.37 mV/W at a magnetic field of +2.26 T. For

negative magnetic fields almost no signal is detected (not shown). In the case

of left-handed circularly polarized light the results are vice versa.

The change of resistance ∆Rx shows the same features (Fig. 26 (a)). Data for

two different measurement techniques are presented: the blue curve in panel

(a) was measured by applying a dc current of either +1 or -1 µA and modulated

THz radiation (method 1, see Section 3.3.3). The largest signal is detected at

1.94 T with ∆Rx = 134.45 Ω. Panel (b) illustrates the second technique.

Herein, the longitudinal resistance Rxx was measured via standard magneto-

transport measurement (see Section 3.3.1) in the dark and in the presence of

unmodulated THz radiation (method 2). These are depicted in black and red,


Figure 25: Measurements of the 8 nm QW square sample #5 obtained

with σ+ polarized radiation of λ = 118.8 µm. The photosignal normalized

by radiation power Ux/P vs. the magnetic field B is shown for (a) T = 4.2 K

and (c) T = 40 K. The right scale in (a) presents the longitudinal resistance

Rxx at 4.2 K. The gray dotted line shows the corresponding calculated CR

absorption profile according to Eq. (45) (see Section 6.2.2). The right scale

in (c) shows the change of photoresistance ∆Rx at 40 K. (b) depicts the

radiation transmission t of the sample in arbitrary units at 4.2 K.

respectively. The difference of the two measured curves corresponds to the

difference of resistance induced by THz radiation, namely the photoresistance

∆Rx. This is shown in Fig. 26 (a), too, as a red line and is in good agreement

with the curve obtained by standard method 1: the period of oscillations is

the same and the largest signal of 183.88 Ω is obtained for the same magnetic

field value of 1.94 T.

When the temperature was increased the oscillations became less pronounced.

For T = 40 K they almost vanish revealing only one dominant peak at 1.84 T

for the photosignal and at 1.86 T for the photoresistance measurement. This is

shown in Fig. 25 (c). These positions coincide very well with the CR position as

determined by radiation transmittance. For right-handed circularly polarized

radiation the transmission decreases by approximately 20% at the CR position


Figure 26: Photoresistance measurement of the 8 nm HgTe QW square

sample #5 at T = 4.2 K and with THz radiation of λ= 118.8 µm. (a) depicts

the change of photoresistance ∆Rx. The blue curve was obtained by method

1 for dc bias and modulated σ+ radiation. The red curve is the difference

of the two curves in (b), presenting the result of method 2 (ac current, un-

modulated THz radiation). (b) shows the magneto-transport measurement

of the longitudinal resistance Rxx with well pronounced Shubnikov-De Haas

oscillations. Rxx was measured with and without unmodulated linearly po-

larized THz radiation (red and black, respectively).

of Bc = +1.8 T (see Fig. 25 (b)). Switching the radiation helicity from σ+ to

σ− cyclotron resonance occurs at Bc = -1.76 T with a decrease of ≈ 22% (not

shown). The cyclotron resonance positions are summarized in Tab. 11 as an

average of the absolute value of Bc for both radiation helicities. The values

are obtained from the high temperature photocurrent and low temperature

radiation transmission measurements.

Similar results were observed for other samples with inverted (#6 and #7) and

normal (#1) parabolic dispersion. The characteristic curves of the photosig-

nal for σ+ radiation at 4.2 and 40 K are shown for these samples in Fig. 27

(a)-(c). The data are accompanied by measurements of the longitudinal resis-


Figure 27: Photosignal normalized by radiation power Ux/P (left scale)

and longitudinal resistance Rxx with SdH oscillations (right scale) for four

different QW widths: (a) 5 nm QW sample #1, (b) 8 nm QW sample #6,

(c) 21 nm QW sample #7 and (d) 6.6 nm QW sample #3. The photosignals

were measured with σ+ polarized radiation of λ = 118.8 µm for two temper-

atures: T = 4.2 K (blue) and T = 40 K (red). THz radiation transmission

is depicted in gray in arbitrary units. The corresponding longitudinal re-

sistance is shown as a black dashed line with the scale on the right. Both,

transmission and longitudinal resistance, were obtained at T = 4.2 K.


tance Rxx (black curves) and radiation transmittance (gray curves) at helium

temperature. All samples show an oscillating behavior of the photosignal at T

= 4.2 K which is reduced to a single peak when the temperature is increased

to T = 40 K. The latter corresponds to the cyclotron resonance position as

identified by radiation transmission measurements. The CR positions Bc are

given in Tab. 11 as an average value of |Bc| for right- and left-handed circularly

polarized light.

6.1.2 Results for a Dirac Fermion System

Finally measurements were carried out for a Dirac fermion system. In Chap-

ter 5 it was shown that the 6.6 nm HgTe QW sample has this special energy

spectrum [14, 15]. Figure 27 (d) shows the magnetic field dependency of the

photocurrent for the ungated 6.6 nm QW square sample #3. As above, the

photosignal was measured for low and high temperature applying right-handed

circularly polarized radiation with λ = 118.8 µm: for T = 4.2 K the photo-

current exhibits slight oscillations which reduce to a single peak when T is

increased. The THz radiation transmission is shown in gray and demonstrates

a clear cyclotron resonance peak at a magnetic field value of B = 1.2 T.

Depending on the carrier density of the QW structure the cyclotron resonance

position Bc shifts (see Chapter 5, compare Eq. (29)). This effect is typical for

systems with a linear dispersion and is caused by the energy dependency of the

effective cyclotron mass mc (Eq. (27)) and by the variation of Fermi energy

(see Chapter 5).

In order to investigate the dependency on the carrier density in more detail

additional measurements were performed on the gated 6.6 nm QW Hall bar

sample #4. In this sample the carrier density n could be exactly varied by

the application of a gate voltage. For this purpose the photosignal Ux was

measured for two static positive magnetic field strengths B = 1.5 and 2 T and

for linearly polarized THz light with λ = 118.8 µm. These measurements were

performed in cooperation with Dimitry Kvon1 and Vasily V. Bel’kov2. The

1Institute of Semiconductor Physics, Novosibirsk, Russia2Ioffe Institute, St. Petersburg, Russia


∝

Figure 28: Photosignal Ux as a function of the carrier density n varied

by the gate voltage of the gated 6.6 nm QW Hall bar structure #4. The

data were obtained at T = 4.2 K and with linear polarized radiation of λ

= 118.8 µm. (a) and (b) depict the photosignal for two different magnetic

fields of B = 2 and 1.5 T, respectively. The black dashed lines show the

longitudinal resistance with SdH oscillations; the gray dotted lines depict

the CR radiation absorption profile calculated according to Eq. (46). (c)

shows the magnetic field value of the maximum photosignal as a function

of the corresponding electron density showing a B ∝ √nc dependency.

experimental results are shown in Fig. 28 as a function of the electron density

n. The photosignal measurements were performed at helium temperature with

THz radiation of 118.8 µm and were accompanied by measurements of the

longitudinal resistance Rxx (left and right scale, respectively). Besides the

observed correlation between the oscillations of the photosignal with those of

the longitudinal resistance Rxx (showing SdH oscillations), the figure indicates

a nonmonotonic behavior of the envelope function with a maximum at carrier


density nc = 19 ×1010 cm−2 for B = 1.5 T and nc = 34 ×1010 cm−2 for B

= 2 T. The same measurements were done for magnetic fields B = 0.5 and

1.85 T with similar results (not shown). The dependency of B on nc is shown

in Fig. 28 (c) for all four static magnetic fields (B = 1, 1.5, 1.85 and 2 T). It

demonstrates a proportionality of B ∝ √nc.

sample LW (nm) BPGEc (T) BTM

c (T)

#1 5 2.66 2.08

#2∗ 8 - -

#3 6.6 1.52 1.16

#4 6.6 - -

#5 8 1.80 1.78

#6 8 1.69 1.90

#7 21 2.18/2.74 2.31/2.91

Table 11: Magnetic field positions Bc of cyclotron resonance for the studied

samples observed in transmission (4.2 K) and high temperature photocur-

rent (40 K) measurements. Listed are the average values of |Bc| for both

helicities. ∗The QW contains cadmium: Hg0.86Cd0.14Te.

6.2 Discussion

The experimental results for the photocurrent and photoresistance measure-

ments of Section 6.1 show a nonmonotonic behavior superimposed by oscil-

lations when an external magnetic field is applied. Moreover, for a 6.6 nm

QW system with a Dirac-like dispersion this behavior could also be observed

as a function of the carrier density. In this section a microscopic picture will

be presented which allows to describe the origin of the THz radiation induced

photocurrents. The model was evolved and calculated by the theoreticians Ma-

rina A. Semina3, Mikhail M. Glazov3 and Leonid E. Golub3 and is based on

the experimental results of this chapter. It is related to the magnetogyrotropic

photogalvanic effect which was introduced and utilized in Chapters 2, 4 and 5.

3Ioffe Institute, St. Petersburg, Russia


Furthermore, it will be shown that the current can be decomposed into two

parts which stem from the de Haas-van Alphen and the Shubnikov-de Haas

effect, respectively. Finally the photoresistance will be addressed. As the

derivation of the equations of this chapter is out of scope of this thesis only

the main theoretical outcome of Ref. [20] will be introduced.

6.2.1 Microscopic Picture of Current Generation

When applying an external magnetic field perpendicular to the plane of a two

dimensional electron gas, due to the Lorentz force the electrons are forced to

move on orbits [39]. Assuming ωcτ ≫ 1 with cyclotron angular frequency ωc

and momentum scattering time τ these orbits can be considered to be closed

with cyclotron radius Rc = v/ωc, where v is the electron velocity [39]. This is

depicted in Fig. 29 for spin-up and -down electrons. Note, that x and y are

arbitrary chosen directions as there are no predefined directions of the currents

due to the low symmetry of the system. Neglecting scattering processes the

electrons move on the solid circles. When electron-phonon scattering is taken

into account the electron energy decreases: the scattering process results in

a displacement of the center of cyclotron orbit by ∆x and in a reduction of

its radius from Rc to R′

c [82, 83]. Hence, the electrons move on the dashed or

dotted circles in Fig. 29 depending on the scattering point shown in the figure

for a positive and negative wavevector ky. In gyrotropic media e.g., HgTe QW

structures, this electron-phonon interaction is spin dependent and given by [20]

Vk′k = V0 + Vzyσz(ky + k′

y). (36)

Here, the first term on the right-hand side V0 corresponds to the conventional

spin independent scattering, V is a second rank pseudotensor, σz is the Pauli

matrix and k (k′) is the initial (scattered) wavevector. Note, that this equation

is a special case of Eq. (10). Considering a fixed spin e.g., spin-up in Fig. 29

(a), Eq. (36) results in unequal probabilities for scattering events for ky < 0

and ky > 0: while the probability is decreased in the first case, it is increased

in the latter case. This results in an electron flux jx,+ in positive x direction of

spin-up electrons. The situation for spin-down electrons is vice versa yielding

a flux jx,− in negative x direction. In the absence of Zeeman splitting these


fluxes are of equal strength and oppositely directed. Therefore, the net electric

current is zero. However, the magnetic field induced Zeeman splitting results

in an unequal population of spin-up and -down subband. Hence, jx,+ and jx,−

do not cancel each other out giving rise to a net dc current jx = jx,+ + jx,−.

The total expression for the dc current is given by [20]

jx =eβζ

|B| |E0|2 [n+µ+(ω)− n−µ−(ω)] , (37)

with β = 1 for parabolic and β = 1/2 for linear dispersion. |E0|2 is the

complex amplitude of the incident radiation, n± is the carrier density and µ±

the electron mobility for each spin branch. ζ stands for the strength of spin

orbit interaction.

The oscillations of the photocurrent observed in experiments enter this picture

as oscillations of carrier densities n± and mobilities µ±. Therefore, it is con-

venient to decompose Eq. (37) into two terms jn and jµ. The former one is

Figure 29: Cyclotron orbits of two dimensional (a) spin-up and (b) spin-

down electrons in an external magnetic field. The solid circles with cyclotron

radius Rc correspond to the cyclotron orbit without electron-phonon scat-

tering. Dashed and dotted circles with cyclotron radius R′c depict cyclotron

orbits for electrons scattered by phonons for two possible scattering events:

ky > 0 and ky < 0. Due to the reduction of Rc by electron-phonon scat-

tering the electrons shift in real space by ∆x = |Rc − R′c|, yielding two

electron fluxes jx,+ and jx,− for spin-up and -down. Note, that x and y are

arbitrarily chosen directions.


related to the spin polarization Sz =n+−n−

2(n++n−)of the system, namely to the dif-

ference in spin-up and -down subband population. This is also known as zero

bias spin separation [5]. The latter contribution originates from the difference

of mobilities µ+(ω)− µ−(ω). Hence, Eq. (37) can be rewritten as

jx = jn + jµ, (38)

with

jn =2eβζ

|B| |E0|2Sznµ(ω) (39)

and

jµ =eβζ

|B| |E0|2nµ+(ω)− µ−(ω)

2. (40)

Herein, n = n+ + n− and µ(ω) = [µ+(ω) + µ−(ω)]/2. In the following these

two contributions will briefly be discussed.

The spin polarization Sz determines jn the first contribution of Eq. (38). It

is dependent on the density of states ν0(E) and originates from the de Haas-

van Alphen effect. Assuming ~ωc ≪ EF and EF ≫ ~

τq, with quantum scatter-

ing time τq, the density of states can be expressed as [82, 84]

ν0(E) =mc

2π~2

[

1± 2exp

( −π

ωcτq

)

cos

(

2πβE

~ωc

)]

. (41)

Herein, ± stands for linear (+) and parabolic (-) dispersion. β is defined as

in Eq. (37). Equation (41) describes the density of states without taking into

account the Zeeman splitting of bands. For spin splitted Landau Levels the

density of states is given by ν±(E) = ν0(E∓∆EZ/2) for each spin branch. This

is sketched in Fig. 4 (b). In Eq. (41) the cosine factor represents the oscillatory

parts and exp(

−πωcτq

)

determines the amplitude of oscillations. Equation (41)

only considers oscillating contributions of the first order term in the small

parameter exp(

−πωcτq

)

- higher order terms are neglected.

The spin polarization SZ [85] can be written in terms of Eq. (41) as

SZ = − 1

4βEF

[

∆EZ±

2~ωc

πT1 exp

( −π

ωcτq

)

sin

(

π∆EZ

~ωc

)

cos

(

2πβEF

~ωc

)]

, (42)


with ± for linear (+) and parabolic (-) dispersion. Here again the cosine fac-

tor represents the oscillatory part and exp(

−πωcτq

)

the amplitude of oscillations.

The sine factor takes into account the relative positions of the Zeeman split

Landau levels and the factor T1 considers the smearing of the carrier distribu-

tion function at nonzero temperatures by [86]

T1 =2π2kBTe

~ωc sinh (2π2kBTe/~ωc), (43)

where Te is the electron gas temperature.

The second term jµ of Eq. (38) is defined by the difference in carrier mobilities

of the spin branches and is related to the Shubnikov-de Haas effect. In line

with Refs. [87,88] the mobility for frequency ω in the vicinity of ωc is given by

µ±(ω) =|e|τ/2mc

1 + (ω − ωc)2τ 2×[

1±

T1 exp

( −π

ωcτq

)

cos

(

2πβE±

~ωc

)

f1(ωτ, ωcτ)

]

, (44)

where E± = EF ∓∆EZ/2 is the Fermi energy in each spin subband, ± stands

for linear (-) and parabolic (+) dispersion and f1 is a smooth function of ωτ

and ωcτ . The factor T1 and β are the same as above (see Eqs. (43) and (37)).

Equation (44) contains both the classical cyclotron resonance part, which is

proportional to τ1+(ω−ωc)2τ2

, as well as the oscillating part, which originates from

the consecutive crossing of Fermi energy and Landau levels (see Section 2.5).

6.2.2 Analysis of the Photocurrents

The theoretical equations of Section 6.2.1 allow a comparison of calculated

photocurrents with measured ones presented in Section 6.1. In order to analyze

the oscillations of the photocurrent the individual contributions jn and jµ are

calculated using Eqs. (39) and (40) with oscillatory parts according to Eqs. (42)

and (44). For this the parameters are chosen close to those determined in

the experiments. In the following three samples will be analyzed which are

typical for normal parabolic, inverted parabolic and linear dispersion: the

8 nm HgCdTe QW Hall bar sample #2, the 8 nm HgTe QW square sample

#5 and the 6.6 nm HgTe QW Hall bar sample #4.


Figure 30: Panel (a) and (b) depict the individual contributions jn and

jµ of the total photocurrent which is presented in (c). The parameters

are chosen close to those of the 8 nm HgCdTe QW Hall bar sample #2,

see Fig. 24: m/m0 = 0.034, n = 75 ×1010 cm−2, g = -36.5 and τq =

0.13 ps. Curves 1, 2 and 3 are calculated for τ = 0.74, 0.5 and 0.3 ps,

respectively. Gray dotted curves correspond to the CR absorption profile

calculated according to Eq. (45).

Calculations for a Parabolic Dispersion Figure 30 (a) and (b) show the two

calculated photocurrent contributions jn and jµ for the 8 nm QW Hall bar

sample #2 with a normal parabolic band structure. The total photocurrent

j = jn + jµ is presented in (c). Curves 1, 2 and 3 correspond to different

transport scattering times τ , namely 0.74, 0.5 and 0.3 ps, respectively. Note,


that all curves are given in arbitrary units and are normalized in the same

way. The gray lines depict the corresponding cyclotron resonance absorption

profiles calculated according to

CR ∝ τ

1 + (ω − ωc)2τ 2. (45)

It is also given in arbitrary units.

Panel (a) and (b) demonstrate that the two contributions jn and jµ have a sig-

nificantly different behavior on the magnetic field, which makes it possible to

distinguish between the two contributing terms: jn mainly follows the smooth

CR absorption profile with a distinct maximum almost at CR position Bc (see

Fig. 30 (a)). Outside the resonance position it decreases strongly, demonstrat-

ing only marginal oscillations. This is in agreement with Eqs. (32) and (33)

of Chapter 5. By contrast, jµ strongly oscillates with several changes of sign.

These can even be observed for magnetic fields larger than Bc. Furthermore,

jµ vanishes at the CR position (see Fig. 30 (b)).

A comparison of the total photocurrent (see Fig. 30 (c)) with the measured

photosignal (see in Fig. 24 (a)) shows that the main features are reproduced.

Both figures show pronounced oscillations with sign inversions of the current

(jµ). The oscillations are enhanced in the range of cyclotron resonance (jn),

but are also observable for B > Bc (jµ). Thus, both terms contribute to the

total current, although jµ seems to dominate.

The same calculations were performed with adjusted parameters for the 8 nm

HgTe QW square sample #5, which has an inverted parabolic band structure.

The results for jn and jµ are presented in Fig. 31. Herein, jn also has a maxi-

mum near Bc and decreases rapidly outside the vicinity of CR. The calculations

reveal no oscillations for jn. On the other hand jµ is zero at CR position and

exhibits oscillations in the range of the CR absorption profile.

A comparison of Fig. 31 for jn and jµ with the experimental findings depicted in

Fig. 25 (a) clearly shows that both terms contribute: the measured signal shows

pronounced oscillations with sign inversions (jµ) as well as an enhancement in

the vicinity of CR (jn).

When the temperature is increased the oscillatory contributions of Eqs. (42)

and (44) are suppressed. This is due to the drastic dependency of T1 on the


Figure 31: Panel (a) and (b) depict the individual contributions jn and

jµ of the total photocurrent. The parameters are chosen close to those of

the 8 nm HgTe QW square sample #5, see Fig. 25: m/m0 = 0.02, n = 24

×1010 cm−2, g = -41.5 and τ = 0.68 ps, τq = 0.13 ps. Gray dotted curves

correspond to the CR absorption profile calculated according to Eq. (45).

effective temperature Te: when Te is increased T1 decreases exponentially by

the sinh(Te) term in the denominator. Hence, only the smooth part of jn

contributes to the total current for high temperatures. This is nicely observed

in the experiment (see Fig. 25 (c)).

It should be noted that the total calculated current j = jn + jµ did not take

into account other current generation mechanism e.g., the spin dependent ex-

citation mechanism [5]. Neither are higher order terms of the small parameter

exp(

−πωcτq

)

considered even though the Shubnikov-de Haas oscillations of Rxx

in Figs. 24 and 25 clearly reveal traces of the second harmonic. Hence, for a

more detailed comparison of theory and experiment these additional mecha-

nism and higher order terms have to be included in the theoretical description.


Figure 32: Panel (a) and (b) depict the individual contributions jn and jµ

of the total photocurrent as a function of the carrier density n, varied by a

gate voltage. The parameters are related to those of the studied 6.6 nm QW

Hall bar structure #4: vDF = 7.6 ×105 m/s (close to the value obtained

experimentally in Chapter 5), τ = 0.8 ps and τq = 0.41 ps for n = 10

×1010 cm−2. Gray dotted curves correspond to the CR absorption profile

calculated with Eq. (46).

Calculations for a Dirac Fermion System A distinguishing feature of CR as-

sisted photocurrents in structures with a linear energy spectrum compared to

a parabolic one is that CR related effects can also be detected upon carrier

density variation. This has its origin in the dependency of the cyclotron fre-

quency on the Fermi energy EF (compare Section 5.3). It can be included into

the calculations of jn and jµ by considering the carrier density dependency of

the conductivity σ(ω), given by [20]

σ(ω) =e2n∗τ ∗

2m∗c

[

1 +(

ω − ω∗c

√

n∗/n)2

τ ∗2n∗/n

] (46)

which is valid in the vicinity of cyclotron resonance. Herein, the parameters

ω∗

c , m∗

c , and τ ∗ are the corresponding values for a certain carrier density n∗.


Figure 32 depicts the theoretical calculations of the photocurrent contributions

for a Dirac-like dispersion as a function of the carrier density n. The used

parameters are similar to those of the 6.6 nm QW Hall bar sample #4 for a

magnetic field B = 2 T. The calculated contribution jn in panel (a) shows a

distinct maximum almost at CR position and gentle oscillations. The second

contribution jµ in panel (b) shows a more pronounced oscillating behavior

than jn with a smooth enhancement, mainly for negative currents, close to the

position of CR. When calculating the two current contributions for a static

magnetic field of 1.5 T the position of the distinct maximum of jn shifts to

smaller carrier densities (not shown). For jµ the enhancement of the oscillations

is shifted to smaller carrier densities, too.

This dependency is also observed in the experiments as shown in Fig. 28. For

the measured curve at B = 2 T the maximum photosignal is detected for a

carrier density of nc = 34 ×1010 cm−2. For 1.5 T the maximum position shifts

to nc = 19 ×1010 cm−2. Figure 28 (c) shows the relationship B(nc) for all

applied magnetic fields B. The data show a B ∝ √nc dependency, which is in

agreement with Eq. (29) for systems with a linear dispersion. Moreover, Fig. 28

(a) and (b) shows that the measured photosignals can nicely be enveloped by

the calculated CR absorption profiles (gray lines, calculated by using Eq. (46)).

6.2.3 Analysis of the Photoresistance

Finally the oscillations in the photoresistance will be addressed, which are

related to the Shubnikov-de Haas effect. Its microscopic origin can be found in

the electron gas heating due to the absorption of THz radiation which causes a

change of the electron mobility (compare Section 2.3). The electron gas can be

described by an effective temperature Te. At cyclotron resonance conditions

the resonant absorption leads to a heating of the electron gas and to an increase

of Te according to

Te = T0 +a

1 + (ω − ωc)2τ 2, (47)

where a is proportional to the radiation intensity and T0 is the lattice temper-

ature. The longitudinal resistance of QWs with a parabolic energy spectrum


is given by the standard expression [86]

Rxx ∝ 2T1 exp (−π

ωcτq) cos(2π

EF

~ωc

) cos(πEZ

~ωc

) +

T2 exp (−2π

ωcτq) cos(4π

EF

~ωc

) cos(2πEZ

~ωc

).(48)

Herein, the first and second term on the right-hand side correspond to the first

and second order term in the small parameter exp(

−πωcτq

)

. Tl with l = 1, 2 is

given by

Tl =2lπ2kBTe

~ωc sinh (2lπ2kBTe/~ωc). (49)

and takes into account the effective temperature Te of the electron gas. Hence,

the Shubnikov-de Haas oscillations smear out when the electron gas is heated

as Tl exponentially decreases with increasing Te.

Figure 33 (a) shows the calculated longitudinal resistance Rxx according to

Eq. (48) for the 8 nm HgTe QW square sample #5. Three different effective

temperatures Te were considered. The suppression of the SdH oscillations can

nicely be seen in the vicinity of CR position: the oscillations for the lowest

effective temperature Te,3 have the largest amplitude; these amplitudes get

notably smaller for higher temperatures Te,2 and Te,1.

Figure 33 (b) shows the calculated photoresistance ∆Rx according to ∆Rx =

Rxx(Te) − Rxx(T0) i.e., the difference of the resistance Rxx(Te,i) for higher

effective temperatures (i = 1, 2, 3) and the dark resistance Rxx(T0) (curve not

shown). ∆Rx shows pronounced oscillations in the vicinity of CR position with

multiple sign inversions.

The corresponding experimental result for the photoresistance ∆Rx is pre-

sented in Figure 26 (a). The comparison shows that the main experimental

features are well described by the presented model: both show pronounced os-

cillations with multiple sign inversions and an enhancement of the signal near

the CR position.

For high temperatures the oscillations in the photoresistance vanish completely.

This is caused by the strong dependency of Tl on the temperature: when Te

is increased, Tl decreases drastically as the denominator contains a sinh(Te)


Figure 33: (a) Calculated longitudinal resistance Rxx for three different

effective electron gas temperatures Te according to Eqs. (47) - (49). Param-

eters are chosen close to the values of the 8 nm HgTe QW square sample

#5: m/m0 = 0.026, n = 25 ×1010 cm−2, g = -41.5, τ = 0.86 ps and τq =

0.16 ps. (b) Photoresistance ∆Rx, described as the difference of the dark

longitudinal resistance Rxx(T0) and the Rxx(Te) curves shown in panel (a).

Gray curves correspond to the CR absorption profile calculated according

to Eq. (45).

term. Therefore, at high temperatures the photoresistance is only caused by

the electron gas heating due to cyclotron resonance. This can be seen nicely in

Fig. 25 (c) for T = 40 K, where the photoresistance just follows the Lorentzian

shape of CR absorption.

6.3 Brief Summary

In this chapter it was shown that THz radiation under cyclotron resonance

conditions can result in quantum oscillations of the photocurrent and the pho-

toresistance at low temperatures. The oscillations were observed as a function

of an external applied magnetic field for all three types of dispersions all re-


alized in HgTe QWs: normal and inverted parabolic, as well as linear. A

microscopic model was developed which explains the photocurrent generation.

It is based on the spin dependent asymmetric electron-phonon scattering. Ac-

cording to the model the current can be decomposed into two contributions.

One is caused by the magnetic field induced electron spin polarization Sz,

analog to the de Haas-van Alphen effect, the second one originates from the

difference of the electron mobilities µ(ω) in the spin subbands in the presence

of a magnetic field. The latter one is related to the Shubnikov-de Haas ef-

fect. Both contain oscillatory parts contributing to the total photocurrent and

describe all experimental observations well.

7 Conclusion

The research carried out in the framework of this thesis clearly demonstrates

that in mercury telluride quantum well structures terahertz radiation induced

photocurrents can be observed and that they are well suited for the charac-

terization of such systems. These currents are of large amplitude and exhibit

different characteristics, comprising oscillations and a strong enhancement if

an external magnetic field is applied. It is shown that the currents originate

from spin dependent phenomena: the terahertz radiation heats the two dimen-

sional electron (hole) gas of the semiconductor structures. The relaxation of

the free carriers is spin dependent and results in an asymmetry of carriers in

momentum space, which yields pure spin currents. Due to the application of an

external magnetic field, these spin currents can be converted into a net electric

current. When the magnetic field is applied perpendicular to the plane of the

two dimensional electron (hole) gas, the currents show a strong increase. It is

demonstrated that this increase is not solely an enhancement due to resonant

absorption of terahertz radiation when the condition for cyclotron resonance is

fulfilled, but also due to material properties like strong spin orbit interaction

and large Lande factor.

Another important result is that the terahertz radiation induced photocurrents

are not only observed for systems with a parabolic energy spectrum, but also

for those with a linear Dirac-like one. Herein, the currents show an unique

behavior: the effect of cyclotron resonance can be detected upon variation of

the carrier density; a phenomenon which is not observable for systems with a

parabolic dispersion. This is attributed to the special Landau level splitting

in these types of systems. In contrast to conventional semiconductors the

Landau level splitting of a system with linear dispersion is not equidistant. As

a consequence, the position where the cyclotron resonance condition is fulfilled

shows a square root dependency on the carrier density. Hence, the cyclotron

resonance assisted currents can be utilized to probe the energy spectrum of

two dimensional Dirac fermions. Moreover, as the currents originate from spin

dependent effects, they give an access to the electronic and spin properties of

these systems e.g., spin polarized electron transport.

7 CONCLUSION 91

All this demonstrates that cyclotron resonance assisted photocurrents are a

suitable method for the investigation of topological insulator systems. There-

fore, it might be possible to use these currents to distinguish between the Dirac

and the insulating states of such systems.

The resonance positions of the mercury telluride quantum well structures of

critical width are observed at rather small magnetic fields. This is a con-

sequence of the extraordinary small cyclotron masses of the Dirac fermion

system. Hence, these mercury telluride quantum well structures are a promis-

ing candidate for the application of frequency selective cyclotron resonance

assisted detectors operating in the terahertz range [89].

In structures with high carrier mobility and at low temperatures the photo-

current exhibits magnetic field induced oscillations. Their analysis shows that

the current can be decomposed into two contributions: one stems from the

periodic change of the carriers’ mobility when the magnetic field is varied, the

second one results from the oscillations of the total spin polarization of the

system. They originate from the consecutive crossing of the Fermi energy with

Landau levels when the magnetic field is swept. They are related to the effects

of Shubnikov-de Haas and de Haas-van Alphen. An interesting feature of the

two current contributions is their behavior at cyclotron resonance position:

whereas the first contribution (the Shubnikov-de Haas related one) vanishes at

the position of cyclotron resonance, the second one (de Haas-van Alphen) is

enhanced.

Last, but not least, the experiments on diluted magnetic semiconductor struc-

tures demonstrate the main properties of the magnetogyrotropic photogalvanic

effect and provide a solid basis for the analysis of the data observed in the mer-

cury telluride quantum well nanostructures.

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Acknowledgment

At this point I would like to thank all those who accompanied and supported

me during my graduation.

In particular, I thank Sergey Ganichev, who gave me the opportunity to work

in his team, who entrusted me with this great and promising topic and who

gave me the possibility to contribute to several, interesting conferences. He

always had a friendly ear for questions and always provided helpful advice

concerning experimental as well as theoretical problems.

Moreover, I would like to thank my colleagues Sergey Danilov and Peter Ol-

brich for all their support and assistance over the last few years. Sincere thanks

are also given to Dima Kvon, Vasily Bel’kov and Dima Kozlov for their exper-

imental advice and know-how, as well as to Grigory Budkin, Marina Semina,

Leonid Golub and in particular to Misha Glazov and Sergey Tarasenko for the

theoretical support and many discussions.

I would also like to thank the whole team for the great working atmosphere

and good team play: Christoph Drexler (thx for the proofreading), Tobias Her-

rmann, Helene Plank and Jakob Munzert and of course all alumni (primarily

Ines Caspers, Thomas Stangl, Sebastian Stachel, Florian Ruckerl and Bruno

Jentzsch), as well as my diploma students Kathrin Dantscher, Philipp Falter-

meier, Patricia Vierling and Peter Lutz. For technical and formal support I

also thank Anton Humbs and Hannelore Lanz.

I thank my parents who enabled my academic studies in the first place and

my whole family for all the backup across the years. In particular, I thank my

dad for the great proofreading and the many (long) discussions.

Last, but not least, I thank Jorg for his detailed proofs and that he always

went with me and jollied me along :)

Documents

Terahertz Laser Radiation Induced Opto-Electronic Eﬀects ... · 2.1 Band Structure of Mercury Telluride HgTe belongs to the group of II-VI semiconductors and crystallizes in zinc-blende